475 Sentences With "functor" | Random Sentence Generator

A bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, the Hom functor is of the type . It can be seen as a functor in two arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other.

This functor is left adjoint to the forgetful functor from groups to sets.

In category theory, a faithful functor (respectively a full functor) is a functor that is injective (respectively surjective) when restricted to each set of morphisms that have a given source and target.

In homological algebra, the relationship between currying and uncurrying is known as tensor-hom adjunction. Here, an interesting twist arises: the Hom functor and the tensor product functor might not lift to an exact sequence; this leads to the definition of the Ext functor and the Tor functor.

The vanishing cycle functor then sits in a distinguished triangle with the nearby cycle functor and a more elementary functor. This formulation has been of continuing influence, in particular in D-module theory.

In the language of category theory, the final topology construction can be described as follows. Let Y be a functor from a discrete category J to the category of topological spaces Top that selects the spaces Yi for i in J. Let Δ be the diagonal functor from Top to the functor category TopJ (this functor sends each space X to the constant functor to X). The comma category (Y ↓ Δ) is then the category of cones from Y, i.e. objects in (Y ↓ Δ) are pairs (X, f) where fi : Yi -> X is a family of continuous maps to X. If U is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to SetJ then the comma category (UY ↓ Δ′) is the category of all cones from UY. The final topology construction can then be described as a functor from (UY ↓ Δ′) to (Y ↓ Δ). This functor is left adjoint to the corresponding forgetful functor.

Cartesian categories with an internal Hom functor that is an adjoint functor to the product are called Cartesian closed categories.

There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom- sets (i.e. it is a full functor). Every functor F : C → D determines a congruence on C by saying f ~ g iff F(f) = F(g).

The universal functor of a diagram is the diagonal functor; its right adjoint is the limit of the diagram and its left adjoint is the colimit. The natural transformation from the diagonal functor to some arbitrary diagram is called a cone.

The free category on a quiver can be described up to isomorphism by a universal property. Let : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let be the forgetful functor defined above, and let be any quiver. Then there is a graph homomorphism : → (()) and given any category D and any graph homomorphism : → , there is a unique functor : () → D such that ()∘=, i.e. the following diagram commutes: 300px The functor is left adjoint to the forgetful functor .

One then calls F an analytic functor, and says that "the Taylor tower converges to the functor", in analogy with Taylor series of an analytic function.

The functor F then factors through the quotient functor C -> C/~ in a unique manner. This may be regarded as the "first isomorphism theorem" for functors.

That is, there is a forgetful functor from Cat to Quiv. Its left adjoint is a free functor which, from a quiver, makes the corresponding free category.

The Hom functor \hom(X,-) commutes with arbitrary limits, while the tensor product -\otimes X functor commutes with arbitrary colimits that exist in their domain category. However, in general, \hom(X,-) fails to commute with colimits, and -\otimes X fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.

The forgetful functors in algebra, such as from Grp to Set, are conservative. More generally, every monadic functor is conservative. In contrast, the forgetful functor from Top to Set is not conservative because not every continuous bijection is a homeomorphism. Every faithful functor from a balanced category is conservative.

A functor is exact if and only if it is both left exact and right exact. A covariant (not necessarily additive) functor is left exact if and only if it turns finite limits into limits; a covariant functor is right exact if and only if it turns finite colimits into colimits; a contravariant functor is left exact if and only if it turns finite colimits into limits; a contravariant functor is right exact if and only if it turns finite limits into colimits. The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functors. Left and right exact functors are ubiquitous mainly because of the following fact: if the functor F is left adjoint to G, then F is right exact and G is left exact.

A local Noetherian ring is regular if and only if its global dimension is finite, say n, which means that any finitely generated R-module has a resolution by projective modules of length at most n. The proof of this and other related statements relies on the usage of homological methods, such as the Ext functor. This functor is the derived functor of the functor :HomR(M, −). The latter functor is exact if M is projective, but not otherwise: for a surjective map E -> F of R-modules, a map M -> F need not extend to a map M -> E. The higher Ext functors measure the non-exactness of the Hom-functor.

In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.

Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits. Let J and C be categories with J a small index category and let C^J be the corresponding functor category. The diagonal functor :\Delta: C \to C^J is the functor that maps each object N in C to the constant functor \Delta(N): J \to C to N (i.e.

In category theory, a branch of mathematics, a fiber functor is a faithful k-linear tensor functor from a tensor category to the category of finite- dimensional k-vector spaces.

The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the overall homology functor for the theory.

The category of small categories Cat has a forgetful functor into the quiver category Quiv: : : Cat → Quiv which takes objects to vertices and morphisms to arrows. Intuitively, "[forgets] which arrows are composites and which are identities". This forgetful functor is right adjoint to the functor sending a quiver to the corresponding free category.

Let R be a ring and M a left R-module. The functor HomR(M,-): Mod-R → Ab is right adjoint to the tensor product functor - \otimesR M: Ab → Mod-R.

Fix a Weil cohomology theory H. It gives a functor from Mnum (pure motives using numerical equivalence) to finite- dimensional \Q-vector spaces. It can be shown that the former category is a Tannakian category. Assuming the equivalence of homological and numerical equivalence, i.e. the above standard conjecture D, the functor H is an exact faithful tensor-functor.

In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by and . Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.

From the point of view of category theory, restriction is an instance of a forgetful functor. This functor is exact, and its left adjoint functor is called induction. The relation between restriction and induction in various contexts is called the Frobenius reciprocity. Taken together, the operations of induction and restriction form a powerful set of tools for analyzing representations.

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.

We can then say that is a subset of if and only if logically implies : the "semantics functor" and the "syntax functor" form a monotone Galois connection, with semantics being the lower adjoint.

Some authors also define a localization of a category C to be an idempotent and coaugmented functor. A coaugmented functor is a pair (L,l) where L:C → C is an endofunctor and l:Id → L is a natural transformation from the identity functor to L (called the coaugmentation). A coaugmented functor is idempotent if, for every X, both maps L(lX),lL(X):L(X) → LL(X) are isomorphisms. It can be proven that in this case, both maps are equal.

Similarly, one can define a colimit as the left adjoint to the diagonal functor given above. To define a homotopy colimit, we must modify in a different way. A homotopy colimit can be defined as the left adjoint to a functor where :, where is the opposite category of . Although this is not the same as the functor above, it does share the property that if the geometric realization of the nerve category () is replaced with a point space, we recover the original functor .

Let X be a scheme. Its functor of points is the functor Hom(−,X) : (Affine schemes)op ⟶ Sets sending an affine scheme A to the set of scheme maps A → X.The Stacks Project, 01J5 A scheme is determined up to isomorphism by its functor of points. This is a stronger version of the Yoneda lemma, which says that a X is determined by the map Hom(−,X):Schemesop → Sets. Conversely, a functor F:(Affine schemes)op → Sets is the functor of points of some scheme if and only if F is a sheaf with respect to the Zariski topology on (Affine schemes), and F admits an open cover by affine schemes.

There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to sets. M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U: Grp → Set has a left adjoint given by the composite KF: Set→Mon→Grp, where F is the free functor; this functor assigns to every set S the free group on S.

A logical functor is a functor between toposes that preserves finite limits and power objects. Logical functors preserve the structures that toposes have. In particular, they preserve finite colimits, subobject classifiers, and exponential objects.

In category theory, a branch of mathematics, a conservative functor is a functor F: C \to D such that for any morphism f in C, F(f) being an isomorphism implies that f is an isomorphism.

In mathematics, the Jacquet module is an module used in the study of automorphic representations. The Jacquet functor is the functor that sends a linear representation to its Jacquet module. They are both named after Hervé Jacquet.

Recall that all finite limits and colimits exist in a pre-abelian category. In general category theory, a functor is called left exact if it preserves all finite limits and right exact if it preserves all finite colimits. (A functor is simply exact if it's both left exact and right exact.) In a pre-abelian category, exact functors can be described in particularly simple terms. First, recall that an additive functor is a functor F: C → D between preadditive categories that acts as a group homomorphism on each hom- set.

An R-module P is projective if and only if the covariant functor is an exact functor, where R-Mod is the category of left R-modules and Ab is the category of abelian groups. When the ring R is commutative, Ab is advantageously replaced by R-Mod in the preceding characterization. This functor is always left exact, but, when P is projective, it is also right exact. This means that P is projective if and only if this functor preserves epimorphisms (surjective homomorphisms), or if it preserves finite colimits.

How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a bifunctor. For we should find a morphism .

Note that if is compact then Ind and ind are the same functor.

The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context;Carnap, Rudolf (1937). The Logical Syntax of Language, Routledge & Kegan, pp. 13–14. see function word.

In category theory, an abstract branch of mathematics, a dominant functor is a functor F : C → D in which every object of D is a retract of an object of the form F(x) for some object X of C..

The higher order directed homotopy theory can be developed through cylinder functor and path functor, all constructions and properties being expressed in the setting of categorical algebra^{[1]}. This approach emphasizes the combinatorial role of cubical sets in directed algebraic topology.

The modifier "cohomological" indicates that the δn raise the index on the T. A covariant homological δ-functor between A and B is similarly defined (and generally uses subscripts), but with δn a morphism Tn(M ) → Tn-1(M). The notions of contravariant cohomological δ-functor between A and B and contravariant homological δ-functor between A and B can also be defined by "reversing the arrows" accordingly.

In general, however, an automorphism group functor may not be represented by a scheme.

Therefore, one also talks about tangent vectors. See also: tangent space to a functor.

In functional programming, a functor is a design pattern inspired by the definition from category theory, that allows for a generic type to apply a function inside without changing the structure of the generic type. This idea is encoded in Haskell using type class class Functor f where fmap :: (a -> b) -> f a -> f b with conditions called functor laws fmap id = id fmap (g . h) = (fmap g) . (fmap h) In Scala higher kinded types are used trait Functor[F[_ { def map[A,B](a: F[A])(f: A => B): F[B] } Simple examples of this are Option and collection types.

Thus, for example, the discrete category with just two objects can be used as a diagram or diagonal functor to define a product or coproduct of two objects. Alternately, for a general category C and the discrete category 2, one can consider the functor category C2. The diagrams of 2 in this category are pairs of objects, and the limit of the diagram is the product. The functor from Set to Cat that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects.

In the language of category theory, any universal construction gives rise to a functor; one thus obtains a functor from the category of commutative monoids to the category of abelian groups which sends the commutative monoid M to its Grothendieck group K. This functor is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids. For a commutative monoid M, the map i : M->K is injective if and only if M has the cancellation property, and it is bijective if and only if M is already a group.

A functor G : C → D induces a map from Cone(F) to Cone(GF): if Ψ is a cone from N to F then GΨ is a cone from GN to GF. The functor G is said to preserve the limits of F if (GL, Gφ) is a limit of GF whenever (L, φ) is a limit of F. (Note that if the limit of F does not exist, then G vacuously preserves the limits of F.) A functor G is said to preserve all limits of shape J if it preserves the limits of all diagrams F : J → C. For example, one can say that G preserves products, equalizers, pullbacks, etc. A continuous functor is one that preserves all small limits. One can make analogous definitions for colimits. For instance, a functor G preserves the colimits of F if G(L, φ) is a colimit of GF whenever (L, φ) is a colimit of F. A cocontinuous functor is one that preserves all small colimits.

If Y is a point, then the direct image equals the global sections functor. Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor f!: D(Y) → D(X).

Though rarer in computer science, one can use category theory directly, which defines a monad as a functor with two additional natural transformations. So to begin, a structure requires a higher-order function (or "functional") named map to qualify as a functor: This is not always a major issue, however, especially when a monad is derived from a pre-existing functor, whereupon the monad inherits automatically. (For historical reasons, this `map` is instead called `fmap` in Haskell.) A monad's first transformation is actually the same from the Kleisli triple, but following the hierarchy of structures closely, it turns out characterizes an applicative functor, an intermediate structure between a monad and a basic functor. In the applicative context, is sometimes referred to as pure but is still the same function.

The use of h^A for the covariant hom-functor and h_A for the contravariant hom-functor is not completely standard. Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors. However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article. The mnemonic "falling into something" can be helpful in remembering that h_A is the contravariant hom-functor.

The Grothendieck group construction is a functor from the category of rings to the category of abelian groups. The higher K-theory should then be a functor from the category of rings but to the category of higher objects such as simplicial abelian groups.

There is no coproduct in Met. The forgetful functor Met → Set assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function. This functor is faithful, and therefore Met is a concrete category.

The terms homological δ-functor and cohomological δ-functor are sometimes used to distinguish between the case where the morphisms "go down" (homological) and the case where they "go up" (cohomological). In particular, one of these modifiers is always implicit, although often left unstated.

Let C be a locally small category and let Set be the category of sets. For each object A of C let Hom(A,-) be the hom functor that maps object X to the set Hom(A,X). A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A,-) for some object A of C. A representation of F is a pair (A, Φ) where :Φ : Hom(A,-) -> F is a natural isomorphism. A contravariant functor G from C to Set is the same thing as a functor G : Cop → Set and is commonly called a presheaf.

In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.

For a given diagram F : J → C and functor G : C → D, if both F and GF have specified limits there is a unique canonical morphism :\tau_F : G \lim F \to \lim GF which respects the corresponding limit cones. The functor G preserves the limits of F if and only this map is an isomorphism. If the categories C and D have all limits of shape J then lim is a functor and the morphisms τF form the components of a natural transformation :\tau:G \lim \to \lim G^J. The functor G preserves all limits of shape J if and only if τ is a natural isomorphism.

Homological algebra began to be studied in its most basic form in the 1800s as a branch of topology, but it wasn't until the 1940s that it became an independent subject with the study of objects such as the ext functor and the tor functor, among others.

A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C. A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : C → D is a full and faithful functor and F(X)\cong F(Y) then X \cong Y.

In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories.

The construction of the free algebra on E is functorial in nature and satisfies an appropriate universal property. The free algebra functor is left adjoint to the forgetful functor from the category of R-algebras to the category of sets. Free algebras over division rings are free ideal rings.

Every limit and colimit provides an example for a simple natural transformation, as a cone amounts to a natural transformation with the diagonal functor as domain. Indeed, if limits and colimits are defined directly in terms of their universal property, they are universal morphisms in a functor category.

In type theory, a polynomial functor (or container functor) is a kind of endofunctor of a category of types that is intimately related to the concept of inductive and coinductive types. Specifically, all W-types (resp. M-types) are (isomorphic to) initial algebras (resp. final coalgebras) of such functors.

Considered as its functor of points, a scheme is a functor which is a sheaf of sets for the Zariski topology on the category of commutative rings, and which, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the étale topology. Michael Artin defined an algebraic space as a functor which is a sheaf in the étale topology and which, locally in the étale topology, is an affine scheme.

If C is a complete category, then, by the above existence theorem for limits, a functor G : C → D is continuous if and only if it preserves (small) products and equalizers. Dually, G is cocontinuous if and only if it preserves (small) coproducts and coequalizers. An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors.

A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given by .

The approximating functors are required to be "k-excisive" – such functors are called polynomial functors by analogy with Taylor polynomials – which is a simplifying condition, and roughly means that they are determined by their behavior around k points at a time, or more formally are sheaves on the configuration space of k points in the given space. The difference between the kth and (k-1)st functors is a "homogeneous functor of degree k" (by analogy with homogeneous polynomials), which can be classified. For the functors T_kF to be approximations to the original functor F, the resulting approximation maps F \to T_kF must be n-connected for some number n, meaning that the approximating functor approximates the original functor "in dimension up to n"; this may not occur. Further, if one wishes to reconstruct the original functor, the resulting approximations must be n-connected for n increasing to infinity.

In mathematics, a topological half-exact functor F is a functor from a fixed topological category (for example CW complexes or pointed spaces) to an abelian category (most frequently in applications, category of abelian groups or category of modules over a fixed ring) that has a following property: for each sequence of spaces, of the form: : X → Y → C(f) where C(f) denotes a mapping cone, the sequence: : F(X) → F(Y) → F(C(f)) is exact. If F is a contravariant functor, it is half-exact if for each sequence of spaces as above, the sequence F(C(f)) → F(Y) → F(X) is exact. Homology is an example of a half-exact functor, and cohomology (and generalized cohomology theories) are examples of contravariant half-exact functors. If B is any fibrant topological space, the (representable) functor F(X)=[X,B] is half-exact.

The codensity monad of a functor G: D \to C is a right Kan extension of G along itself.

If there is a monoidal functor from a monoidal category M to a monoidal category N, then any category enriched over M can be reinterpreted as a category enriched over N. Every monoidal category M has a monoidal functor M(I, -) to the category of sets, so any enriched category has an underlying ordinary category. In many examples (such as those above) this functor is faithful, so a category enriched over M can be described as an ordinary category with certain additional structure or properties.

There is a natural functor from Ring to the category of groups, Grp, which sends each ring R to its group of units U(R) and each ring homomorphism to the restriction to U(R). This functor has a left adjoint which sends each group G to the integral group ring Z[G]. Another functor between these categories sends each ring R to the group of units of the matrix ring M2(R) which acts on the projective line over a ring P(R).

A version of the representability theorem in the case of triangulated categories is due to Amnon Neeman. Together with the preceding remark, it gives a criterion for a (covariant) functor F: C → D between triangulated categories satisfying certain technical conditions to have a right adjoint functor. Namely, if C and D are triangulated categories with C compactly generated and F a triangulated functor commuting with arbitrary direct sums, then F is a left adjoint. Neeman has applied this to proving the Grothendieck duality theorem in algebraic geometry.

In homological algebra, a δ-functor between two abelian categories A and B is a collection of functors from A to B together with a collection of morphisms that satisfy properties generalising those of derived functors. A universal δ-functor is a δ-functor satisfying a specific universal property related to extending morphisms beyond "degree 0". These notions were introduced by Alexander Grothendieck in his "Tohoku paper" to provide an appropriate setting for derived functors.Grothendieck 1957 In particular, derived functors are universal δ-functors.

Both of these functors have left adjoints. The left adjoint of A is the functor which assigns to every abelian group X (thought of as a Z-module) the tensor ring T(X). The left adjoint of M is the functor which assigns to every monoid X the integral monoid ring Z[X].

This construction gives rise to a functor \pi_1 from the category of directed spaces to the category of small categories.

In this sense, the functor G can be said to commute with limits (up to a canonical natural isomorphism). Preservation of limits and colimits is a concept that only applies to covariant functors. For contravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.

The passage from a category C to Ind(C) amounts to freely adding filtered colimits to the category. This is why the construction is also referred to as the ind-completion of C. This is made precise by the following assertion: any functor F: C \to D taking values in a category D which has all filtered colimits extends to a functor Ind(C) \to D which is uniquely determined by the requirements that its value on C is the original functor F and such that it preserves all filtered colimits.

If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for isomorphisms in an algebraic setting, the composite of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be naturally isomorphic to its image under this composition.

A functor F \colon C\to D can be seen as a profunctor \phi_F \colon C rightarrow D by postcomposing with the Yoneda functor: :\phi_F=Y_D\circ F. It can be shown that such a profunctor \phi_F has a right adjoint. Moreover, this is a characterization: a profunctor \phi \colon C rightarrow D has a right adjoint if and only if \hat\phi \colon C\to\hat D factors through the Cauchy completion of D, i.e. there exists a functor F \colon C\to D such that \hat\phi=Y_D\circ F.

A multifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctor with .

Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.

This means that T is left adjoint to the forgetful functor U (see the section below on relation to adjoint functors).

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories.

In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.

There are thus no zero objects in Ord. The categorical product in Ord is given by the product order on the cartesian product. We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category.

In category theory, a branch of mathematics, a subfunctor is a special type of functor that is an analogue of a subset.

A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(-,A) for some object A of C.

In this setting, the category of linear representations of over is the functor category → Vect, which has natural transformations as its morphisms.

Equivariant maps can be generalized to arbitrary categories in a straightforward manner. Every group G can be viewed as a category with a single object (morphisms in this category are just the elements of G). Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object. For example, a G-set is equivalent to a functor from G to the category of sets, Set, and a linear representation is equivalent to a functor to the category of vector spaces over a field, VectK. Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.

The étale cohomology groups Hi(F) of the sheaf F of abelian groups are defined as the right derived functors of the functor of sections, :F \to \Gamma(F) (where the space of sections Γ(F) of F is F(X)). The sections of a sheaf can be thought of as Hom(Z, F) where Z is the sheaf that returns the integers as an abelian group. The idea of derived functor here is that the functor of sections doesn't respect exact sequences as it is not right exact; according to general principles of homological algebra there will be a sequence of functors H 0, H 1, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The H 0 functor coincides with the section functor Γ. More generally, a morphism of schemes f : X → Y induces a map f∗ from étale sheaves over X to étale sheaves over Y, and its right derived functors are denoted by Rqf∗, for q a non-negative integer.

The category of rings, Ring, is a nonfull subcategory of Rng. It is nonfull because there are rng homomorphisms between rings which do not preserve the identity, and are therefore not morphisms in Ring. The inclusion functor Ring → Rng has a left adjoint which formally adjoins an identity to any rng. The inclusion functor Ring → Rng respects limits but not colimits.

For small categories, this is the same as the action on objects of the categorical product in the category Cat. A functor whose domain is a product category is known as a bifunctor. An important example is the Hom functor, which has the product of the opposite of some category with the original category as domain: :Hom : Cop × C → Set.

An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure. If C and D are M-categories (that is, categories enriched over monoidal category M), an M-enriched functor T: C → D is a map which assigns to each object of C an object of D and for each pair of objects a and b in C provides a morphism in M Tab : C(a, b) → D(T(a), T(b)) between the hom-objects of C and D (which are objects in M), satisfying enriched versions of the axioms of a functor, viz preservation of identity and composition. Because the hom-objects need not be sets in an enriched category, one cannot speak of a particular morphism.

A strongly ribbon category is a ribbon category C equipped with a dagger structure such that the functor †: Cop → C coherently preserves the ribbon structure.

There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed above.

We choose . This operation on morphisms is called cartesian product of morphisms. Second, consider the general product functor. For families we should find a morphism .

In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.

The tensor product is another non-exact functor relevant in the context of commutative rings: for a general R-module M, the functor :M ⊗R − is only right exact. If it is exact, M is called flat. If R is local, any finitely presented flat module is free of finite rank, thus projective. Despite being defined in terms of homological algebra, flatness has profound geometric implications.

In mathematics, a full subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector, or localization. Dually, A is said to be coreflective in B when the inclusion functor has a right adjoint. Informally, a reflector acts as a kind of completion operation.

The "combinatory" (the word is Quine's) predicate functors, all monadic and peculiar to PFL, are Inv, inv, ∃, +, and p. A term is either an atomic term, or constructed by the following recursive rule. If τ is a term, then Invτ, invτ, ∃τ, +τ, and pτ are terms. A functor with a superscript n, n a natural number > 1, denotes n consecutive applications (iterations) of that functor.

A formula is either a term or defined by the recursive rule: if α and β are formulas, then αβ and ~(α) are likewise formulas. Hence "~" is another monadic functor, and concatenation is the sole dyadic predicate functor. Quine called these functors "alethic." The natural interpretation of "~" is negation; that of concatenation is any connective that, when combined with negation, forms a functionally complete set of connectives.

Freyd obtained his Ph.D. from Princeton University in 1960; his dissertation, on Functor Theory, was written under the supervision of Norman Steenrod and David Buchsbaum. Freyd is best known for his adjoint functor theorem. He was the author of the foundational book Abelian Categories: An Introduction to the Theory of Functors (1964). This work culminates in a proof of the Freyd–Mitchell embedding theorem.

The classical, geometric theory of Solomon Lefschetz was recast in purely algebraic terms, in SGA7. This was for the requirements of its application in the context of l-adic cohomology; and eventual application to the Weil conjectures. There the definition uses derived categories, and looks very different. It involves a functor, the nearby cycle functor, with a definition by means of the higher direct image and pullbacks.

In mathematics, a bivariant theory was introduced by Fulton and MacPherson , in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring. On technical levels, a bivariant theory is a mix of a homology theory and a cohomology theory. In general, a homology theory is a covariant functor from the category of spaces to the category of abelian groups, while a cohomology theory is a contravariant functor from the category of (nice) spaces to the category of rings. A bivariant theory is a functor both covariant and contravariant; hence, the name “bivariant”.

Subfunctors in general are like global versions of subsets. For example, if one imagines the objects of some category C to be analogous to the open sets of a topological space, then a contravariant functor from C to the category of sets gives a set-valued presheaf on C, that is, it associates sets to the objects of C in a way that is compatible with the arrows of C. A subfunctor then associates a subset to each set, again in a compatible way. The most important examples of subfunctors are subfunctors of the Hom functor. Let c be an object of the category C, and consider the functor .

Subfunctors are also used in the construction of representable functors on the category of ringed spaces. Let F be a contravariant functor from the category of ringed spaces to the category of sets, and let G ⊆ F. Suppose that this inclusion morphism G → F is representable by open immersions, i.e., for any representable functor and any morphism , the fibered product is a representable functor and the morphism Y → X defined by the Yoneda lemma is an open immersion. Then G is called an open subfunctor of F. If F is covered by representable open subfunctors, then, under certain conditions, it can be shown that F is representable.

For any topological group G admitting the structure of a CW-complex, one may consider the category of principal G-bundles. One can define a functor from the category of CW-complexes to the category of sets by assigning to each CW-complex X the set of principal G-bundles on X. This functor descends to a functor on the homotopy category of CW-complexes, and it is natural to ask whether the functor so obtained is representable. The answer is affirmative, and the representing object is called the classifying space of the group G and typically denoted BG. If we restrict our attention to the homotopy category of CW-complexes, then BG is unique. Any CW-complex that is homotopy equivalent to BG is called a model for BG. For example, if G is the group of order 2, then a model for BG is infinite-dimensional real projective space.

Formally, a diagram of type J in a category C is a (covariant) functor The category J is called the index category or the scheme of the diagram D; the functor is sometimes called a J-shaped diagram. The actual objects and morphisms in J are largely irrelevant; only the way in which they are interrelated matters. The diagram D is thought of as indexing a collection of objects and morphisms in C patterned on J. Although, technically, there is no difference between an individual diagram and a functor or between a scheme and a category, the change in terminology reflects a change in perspective, just as in the set theoretic case: one fixes the index category, and allows the functor (and, secondarily, the target category) to vary. One is most often interested in the case where the scheme J is a small or even finite category.

A proof from the Grundlagen der Mathematik that a sufficiently strong consistent theory cannot contain its own reference functor is now known as the Hilbert–Bernays paradox.

In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F, with specific properties as defined below. For both algebra and coalgebra, a functor is a convenient and general way of organizing a signature. This has applications in computer science: examples of coalgebras include lazy, infinite data structures, such as streams, and also transition systems. F-coalgebras are dual to F-algebras.

Because Ab has kernels, one can then show that Ab is a complete category. The coproduct in Ab is given by the direct sum; since Ab has cokernels, it follows that Ab is also cocomplete. We have a forgetful functor Ab → Set which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore Ab is a concrete category.

Given two closed model categories C and D, a Quillen adjunction is a pair :(F, G): C \leftrightarrows D of adjoint functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and trivial fibrations. In such an adjunction F is called the left Quillen functor and G is called the right Quillen functor.

Grothendieck's definition of sheaf cohomology, now standard, uses the language of homological algebra. The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups on X to abelian groups. In more detail, start with the functor E ↦ E(X) from sheaves of abelian groups on X to abelian groups. This is left exact, but in general not right exact.

The Grothendieck group is the fundamental construction of K-theory. The group K_0(M) of a compact manifold M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundles of finite rank on M with the monoid operation given by direct sum. This gives a contravariant functor from manifolds to abelian groups. This functor is studied and extended in topological K-theory.

In algebraic topology, a branch of mathematics, the Čech-to-derived functor spectral sequence is a spectral sequence that relates Čech cohomology of a sheaf and sheaf cohomology.

If the functor G admits a left adjoint F, the codensity monad is given by the composite G \circ F, together with the standard unit and multiplication maps.

In mathematics, in the theory of sheaves the direct image with compact (or proper) support is an image functor for sheaves. It is one of Grothendieck's six operations.

Suppose C is a small category (i.e. the objects and morphisms form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D), [C,D], or D ^C, has as objects the covariant functors from C to D, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if \mu (X) : F(X) \to G(X) is a natural transformation from the functor F : C \to D to the functor G : C \to D, and \eta(X) : G(X) \to H(X) is a natural transformation from the functor G to the functor H, then the collection \eta(X)\mu(X) : F(X) \to H(X) defines a natural transformation from F to H. With this composition of natural transformations (known as vertical composition, see natural transformation), D^C satisfies the axioms of a category.

It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived functor theorem of Quillen says that the total left derived functor :LF: Ho(C) -> Ho(D) is a left adjoint to the total right derived functor :RG: Ho(D) -> Ho(C). This adjunction (LF, RG) is called the derived adjunction. If (F, G) is a Quillen adjunction as above such that :F(c) -> d with c cofibrant and d fibrant is a weak equivalence in D if and only if :c -> G(d) is a weak equivalence in C then it is called a Quillen equivalence of the closed model categories C and D. In this case the derived adjunction is an adjoint equivalence of categories so that :LF(c) -> d is an isomorphism in Ho(D) if and only if :c -> RG(d) is an isomorphism in Ho(C).

In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup. The idea is to try to construct a p'-subgroup of a finite group G, which has a good chance of being normal in G, by taking as generators certain p'-subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian p-subgroups of G. The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including who defined signalizer functors, who proved the Solvable Signalizer Functor Theorem for solvable groups, and who proved it for all groups. This theorem is needed to prove the so-called "dichotomy" stating that a given nonabelian finite simple group either has local characteristic two, or is of component type.

Our F easily extends to a functor Set → BA, and our definition of X generating a free Boolean algebra FX is precisely that U has a left adjoint F.

Then it turns out that a functor between pre-abelian categories is left exact if and only if it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels. Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages. Exact functors are most useful in the study of abelian categories, where they can be applied to exact sequences.

In homological algebra, the hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex. Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories.

The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on \Delta, that is a contravariant functor from \Delta to another category. For instance, simplicial sets are contravariant with the codomain category being the category of sets. A cosimplicial object is defined similarly as a covariant functor originating from \Delta.

The free commutative ring on a set of generators E is the polynomial ring Z[E] whose variables are taken from E. This gives a left adjoint functor to the forgetful functor from CRing to Set. CRing is limit-closed in Ring, which means that limits in CRing are the same as they are in Ring. Colimits, however, are generally different. They can be formed by taking the commutative quotient of colimits in Ring.

The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set considered as a discrete category.

Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take as the discrete category with two objects, so that is simply the product category . The diagonal functor assigns to each object the ordered pair and to each morphism the pair . The product in is given by a universal morphism from the functor to the object in .

The symmetric algebra is a functor from the category of -modules to the category of -commutative algebra, since the universal property implies that every module homomorphism f:V\to W can be uniquely extended to an algebra homomorphism S(f):S(V)\to S(W). The universal property can be reformulated by saying that the symmetric algebra is a left adjoint to the forgetful functor that sends a commutative algebra to its underlying module.

This is the internal hom [x, y]. Every closed category has a forgetful functor to the category of sets, which in particular takes the internal hom to the external hom.

The entire proof turned on the existence of a right adjoint to a certain functor. This is something undeniably abstract, and non- constructive, but also powerful in its own way.

In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.

In topology, a branch of mathematics, a cubical set is a set-valued contravariant functor on the category of (various) n-cubes. See the references for the more precise definitions.

The idea of homotopy can be turned into a formal category of category theory. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.

The category of rings is, therefore, isomorphic to the category Z-Alg.. Many statements about the category of rings can be generalized to statements about the category of R-algebras. For each commutative ring R there is a functor R-Alg → Ring which forgets the R-module structure. This functor has a left adjoint which sends each ring A to the tensor product R⊗ZA, thought of as an R-algebra by setting r·(s⊗a) = rs⊗a.

Other types of forgetfulness also give rise to objects quite like free objects, in that they are left adjoint to a forgetful functor, not necessarily to sets. For example, the tensor algebra construction on a vector space is the left adjoint to the functor on associative algebras that ignores the algebra structure. It is therefore often also called a free algebra. Likewise the symmetric algebra and exterior algebra are free symmetric and anti-symmetric algebras on a vector space.

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module by another.

For any locally compact abelian (LCA) group A, the group of continuous homomorphisms :Hom(A, S1) from A to the circle group is again locally compact. Pontryagin duality asserts that this functor induces an equivalence of categories :LCAop -> LCA. This functor exchanges several properties of topological groups. For example, finite groups correspond to finite groups, compact groups correspond to discrete groups, and metrisable groups correspond to countable unions of compact groups (and vice versa in all statements).

250px In mathematics, the Stone functor is a functor S: Topop → Bool, where Top is the category of topological spaces and Bool is the category of Boolean algebras and Boolean homomorphisms. It assigns to each topological space X the Boolean algebra S(X) of its clopen subsets, and to each morphism fop: X → Y in Topop (i.e., a continuous map f: Y → X) the homomorphism S(f): S(X) → S(Y) given by S(f)(Z) = f−1[Z].

In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck.

A neutral Tannakian category is a rigid abelian tensor category, such that there exists a K-tensor functor to the category of finite dimensional K-vector spaces that is exact and faithful.

This functor has a left adjoint (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to that set equipped with the total relation).

For a space X, pt(Ω(X)) is called its soberification. The case of the functor Ω o pt is symmetric but a special name for this operation is not commonly used.

A geometric morphism (u∗,u∗) is essential if u∗ has a further left adjoint u!, or equivalently (by the adjoint functor theorem) if u∗ preserves not only finite but all small limits.

Given some category T of topological spaces (possibly with some additional structure) such as the category of all topological spaces Top or the category of pointed topological spaces, that is, topological spaces with a distinguished base point, and a functor F: T \to A from that category into some category A of algebraic structures such as the category of groups Grp or of abelian groups Ab which then associates such an algebraic structure to every topological space, then for every morphism f: X \to Y of T (which is usually a continuous map, possibly preserving some other structure such as the base point) this functor induces an induced morphism F(f): F(X) \to F(Y) in A (which is a group homomorphism if A is a category of groups) between the algebraic structures F(X) and F(Y) associated to X and Y, respectively. If F is not a functor but a contravariant functor then by definition it induces morphisms in the opposite direction: F(f): F(Y) \to F(X). Cohomology groups give an example.

The inverse functor is defined by realizing that for any Mn(R)-module there is a left R-module X such that the Mn(R)-module is obtained from X as described above.

To motivate the (conjectural) motivic Galois group, fix a field k and consider the functor :finite separable extensions K of k → non- empty finite sets with a (continuous) transitive action of the absolute Galois group of k which maps K to the (finite) set of embeddings of K into an algebraic closure of k. In Galois theory this functor is shown to be an equivalence of categories. Notice that fields are 0-dimensional. Motives of this kind are called Artin motives.

Suppose that R is a Noetherian complete local ring with residue field k, and choose E to be an injective hull of k (sometimes called a Matlis module). The dual DR(M) of a module M is defined to be HomR(M,E). Then Matlis duality states that the duality functor DR gives an anti-equivalence between the categories of Artinian and Noetherian R-modules. In particular the duality functor gives an anti-equivalence from the category of finite-length modules to itself.

The category of commutative rings, denoted CRing, is the full subcategory of Ring whose objects are all commutative rings. This category is one of the central objects of study in the subject of commutative algebra. Any ring can be made commutative by taking the quotient by the ideal generated by all elements of the form (xy − yx). This defines a functor Ring → CRing which is left adjoint to the inclusion functor, so that CRing is a reflective subcategory of Ring.

An indiscrete category is a category C in which every hom-set C(X, Y) is a singleton. Every class X gives rise to an indiscrete category whose objects are the elements of X with exactly one morphism between any two objects. Any two nonempty indiscrete categories are equivalent to each other. The functor from Set to Cat that sends a set to the corresponding indiscrete category is right adjoint to the functor that sends a small category to its set of objects.

Here, the forgetful functor from commutative algebras to vector spaces or modules (forgetting the multiplication) is the composition of the forgetful functors from commutative algebras to associative algebras (forgetting commutativity), and from associative algebras to vectors or modules (forgetting the multiplication). As the tensor algebra and the quotient by commutators are left adjoint to these forgetful functors, their composition is left adjoint to the forgetful functor from commutative algebra to vectors or modules, and this proves the desired universal property.

But it turns out that (if A is "nice" enough) there is one canonical way of doing so, given by the right derived functors of F. For every i≥1, there is a functor RiF: A → B, and the above sequence continues like so: 0 → F(A) → F(B) → F(C) → R1F(A) → R1F(B) → R1F(C) → R2F(A) → R2F(B) → ... . From this we see that F is an exact functor if and only if R1F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact. If the object A in the above short exact sequence is injective, then the sequence splits. Applying any additive functor to a split sequence results in a split sequence, so in particular R1F(A) = 0.

If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab. That is, F is additive if and only if, given any objects A and B of C, the function f: Hom(A,B) → Hom(F(A),F(B)) is a group homomorphism. Most functors studied between preadditive categories are additive. For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a ring homomorphism from R to S is represented by an additive functor from R to S, and conversely. If C and D are categories and D is preadditive, then the functor category DC is also preadditive, because natural transformations can be added in a natural way.

The tensor algebra T(V) is also called the free algebra on the vector space V, and is functorial. As with other free constructions, the functor T is left adjoint to some forgetful functor. In this case, it's the functor that sends each K-algebra to its underlying vector space. Explicitly, the tensor algebra satisfies the following universal property, which formally expresses the statement that it is the most general algebra containing V: : Any linear transformation f : V -> A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T(V) to A as indicated by the following commutative diagram: Universal property of the tensor algebra Here i is the canonical inclusion of V into T(V) (the unit of the adjunction).

An example of a filter rule from the true-analogy rule set creates match hypotheses between predicates that have the same functor. The true- analogy rule set has an intern rule that iterates over the arguments of any match hypothesis, creating more match hypotheses if the arguments are entities or functions, or if the arguments are attributes and have the same functor. In order to illustrate how the match rules produce match hypotheses consider these two predicates: `transmit torque inputgear secondgear (p1)` `transmit signal switch div10 (p2)` Here we use true analogy for the type of reasoning. The filter match rule generates a match between p1 and p2 because they share the same functor, transmit. The intern rules then produce three more match hypotheses: torque to signal, inputgear to switch, and secondgear to div10.

Identifying the limit of Hom(F-, N) with the set Cocone(F, N), this relationship can be used to define the colimit of the diagram F as a representation of the functor Cocone(F, -).

For example, if we take the category of vector spaces, we obtain group representations in this fashion. We can view a group G as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant) functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors.

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes". For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another.

A functor is an operation on spaces and functions between them. It associates to each space a new space and to each function between two spaces a new function between the corresponding new spaces. In each of the above cases, the functor sends each space to its tangent bundle and it sends each function to its derivative. For example, in the manifold case, the derivative sends a Cr-manifold to a Cr−1-manifold (its tangent bundle) and a Cr-function to its total derivative.

In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk(R,M) are isomorphism classes of commutative k-algebras E with a homomorphism onto the k-algebra R whose kernel is the R-module M (with all pairs of elements in M having product 0). Note that some authors use Exal as the same functor. There are similar functors Exal and Exan for non-commutative rings and algebras, and functors Exaltop, Exantop.

If a variable appears in a real or finite domain expression, it can only take a value in the reals or the finite domain. Such a variable cannot take a term made of a functor applied to other terms as a value. The constraint store is unsatisfiable if a variable is bound to take both a value of the specific domain and a functor applied to terms. After a constraint is added to the constraint store, some operations are performed on the constraint store.

If (C, J) and (D, K) are sites and u : C → D is a functor, then u is continuous if for every sheaf F on D with respect to the topology K, the presheaf Fu is a sheaf with respect to the topology J. Continuous functors induce functors between the corresponding topoi by sending a sheaf F to Fu. These functors are called pushforwards. If \tilde C and \tilde D denote the topoi associated to C and D, then the pushforward functor is u_s : \tilde D \to \tilde C. us admits a left adjoint us called the pullback. us need not preserve limits, even finite limits. In the same way, u sends a sieve on an object X of C to a sieve on the object uX of D. A continuous functor sends covering sieves to covering sieves.

Referring to the above commutative diagram, one observes that every morphism :h : A′ -> A gives rise to a natural transformation :Hom(h,-) : Hom(A,-) -> Hom(A′,-) and every morphism :f : B -> B′ gives rise to a natural transformation :Hom(-,f) : Hom(-,B) -> Hom(-,B′) Yoneda's lemma implies that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category SetCop (covariant or contravariant depending on which Hom functor is used).

The Jacquet module J(V) of a representation (π,V) of a group N is the space of co-invariants of N; or in other words the largest quotient of V on which N acts trivially, or the zeroth homology group H0(N,V). In other words, it is the quotient V/VN where VN is the subspace of V generated by elements of the form π(n)v - v for all n in N and all v in V. The Jacquet functor J is the functor taking V to its Jacquet module J(V).

Given a quasi-category C, one can associate to it an ordinary category hC, called the homotopy category of C. The homotopy category has as objects the vertices of C. The morphisms are given by homotopy classes of edges between vertices. Composition is given using the horn filler condition for n = 2\. For a general simplicial set there is a functor \tau_1 from sSet to Cat, known as the fundamental category functor, and for a quasi-category C the fundamental category is the same as the homotopy category, i.e. \tau_1(C)=hC.

The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint. Taking direct limits in Ab is an exact functor. Since the group of integers Z serves as a generator, the category Ab is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category. An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group.

A simplicial object X in a category C is a contravariant functor :X : Δ -> C or equivalently a covariant functor :X: Δop -> C, where Δ still denotes the simplex category. When C is the category of sets, we are just talking about the simplicial sets that were defined above. Letting C be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups, respectively. Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets.

The codensity monad of a functor G: D \to C is defined to be the right Kan extension of G along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor :T^G : C \to C. The monad structure on T^G stems from the universal property of the right Kan extension. The codensity monad exists whenever D is a small category (has only a set, as opposed to a proper class, of morphisms) and C possesses all (small, i.e., set-indexed) limits.

The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified by Aschbacher in 1978 # Groups with a strongly p-embedded subgroup for p odd # The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors. This was solved by McBride in 1982. # Groups of characteristic p type. This is the problem of groups with a strongly p-embedded 2-local subgroup with p odd, which was handled by Aschbacher.

They are contravariant if they change by the inverse transformation. This is sometimes a source of confusion for two distinct but related reasons. The first is that vectors whose components are covariant (called covectors or 1-forms) actually pull back under smooth functions, meaning that the operation assigning the space of covectors to a smooth manifold is actually a contravariant functor. Likewise, vectors whose components are contravariant push forward under smooth mappings, so the operation assigning the space of (contravariant) vectors to a smooth manifold is a covariant functor.

The construction of free groups is a common and illuminating example. Let F : Set → Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G: Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let \eta_Y:Y\to GFY be the set map given by "inclusion of generators".

In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory.

Any functor K : C → Set with a left adjoint F : Set → C is represented by (FX, ηX(•)) where X = {•} is a singleton set and η is the unit of the adjunction. Conversely, if K is represented by a pair (A, u) and all small copowers of A exist in C then K has a left adjoint F which sends each set I to the Ith copower of A. Therefore, if C is a category with all small copowers, a functor K : C → Set is representable if and only if it has a left adjoint.

If X and Y are topoi, a geometric morphism is a pair of adjoint functors (u∗,u∗) (where u∗ : Y → X is left adjoint to u∗ : X → Y) such that u∗ preserves finite limits. Note that u∗ automatically preserves colimits by virtue of having a right adjoint. By Freyd's adjoint functor theorem, to give a geometric morphism X → Y is to give a functor u∗: Y → X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales.

Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category.

In languages such as Lua, more sophisticated techniques exist which allow a function to be replaced by a new function with the same name, which would permit: factorial = construct-memoized- functor(factorial) Essentially, such techniques involve attaching the original function object to the created functor and forwarding calls to the original function being memoized via an alias when a call to the actual function is required (to avoid endless recursion), as illustrated below: function construct-memoized-functor (F is a function object parameter) allocate a function object called memoized-version; let memoized-version(arguments) be if self has no attached array values then [self is a reference to this object] allocate an associative array called values; attach values to self; allocate a new function object called alias; attach alias to self; [for later ability to invoke F indirectly] self.alias = F; end if; if self.values[arguments] is empty then self.values[arguments] = self.

In another group of dualities, the objects of one theory are translated into objects of another theory and the maps between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using the parlance of category theory, this amounts to a contravariant functor between two categories and : which for any two objects X and Y of C gives a map That functor may or may not be an equivalence of categories. There are various situations, where such a functor is an equivalence between the opposite category of , and . Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed.. Therefore, any duality between categories and is formally the same as an equivalence between and ( and ).

100px : Let X be a set and i\colon X\to L a morphism of sets from X into a Lie algebra L. The Lie algebra L is called free on X if i is the universal morphism; that is, if for any Lie algebra A with a morphism of sets f\colon X \to A, there is a unique Lie algebra morphism g\colon L\to A such that f = g\circ i. Given a set X, one can show that there exists a unique free Lie algebra L(X) generated by X. In the language of category theory, the functor sending a set X to the Lie algebra generated by X is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functor. The free Lie algebra on a set X is naturally graded.

The analogous concept in category theory is called a diagram. A diagram is a functor giving rise to an indexed family of objects in a category C, indexed by another category J, and related by morphisms depending on two indices.

These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where homology is expressible as a functor from the category of topological spaces to the category of graded abelian groups.

Many investigations around this problem involve diagrams of semilattices or of algebras. A most useful folklore result about these is the following. Theorem. The functor Conc, defined on all algebras of a given signature, to all (∨,0)-semilattices, preserves direct limits.

Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor :U : Top -> Set to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function. The forgetful functor U has both a left adjoint :D : Set -> Top which equips a given set with the discrete topology, and a right adjoint :I : Set -> Top which equips a given set with the indiscrete topology.

The augmented simplex category, denoted by \Delta_+ is the category of all finite ordinals and order-preserving maps, thus \Delta_+=\Delta\cup [-1], where [-1]=\emptyset. Accordingly, this category might also be denoted FinOrd. The augmented simplex category is occasionally referred to as algebraists' simplex category and the above version is called topologists' simplex category. A contravariant functor defined on \Delta_+ is called an augmented simplicial object and a covariant functor out of \Delta_+ is called an augmented cosimplicial object; when the codomain category is the category of sets, for example, these are called augmented simplicial sets and augmented cosimplicial sets respectively.

A topological quantum field theory is a monoidal functor from a category of cobordisms to a category of vector spaces. That is, it is a functor whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds. In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not. For instance, the disk bounding the circle corresponds to a null-ary operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below). This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e.

The long list of examples in this article indicates that common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve colimits/limits (which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

The idea of adjoint functors was introduced by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as :hom(F(X), Y) = hom(X, G(Y)) in the category of abelian groups, where F was the functor \- \otimes A (i.e. take the tensor product with A), and G was the functor hom(A,–) (this is now known as the tensor-hom adjunction).

Formally, given two categories C and D, an equivalence of categories consists of a functor F : C → D, a functor G : D → C, and two natural isomorphisms ε: FG→ID and η : IC→GF. Here FG: D→D and GF: C→C, denote the respective compositions of F and G, and IC: C→C and ID: D→D denote the identity functors on C and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality of categories instead. One often does not specify all the above data.

The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). Sending each module M to the group of invariants M^G yields a functor from the category of G-modules to the category Ab of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors.This uses that the category of G-modules has enough injectives, since it is isomorphic to the category of all modules over the group ring \Z[G].

The Karoubi envelope construction associates to an arbitrary category C a category kar(C) together with a functor :s:C\rightarrow kar(C) such that the image s(p) of every idempotent p in C splits in kar(C). When applied to a preadditive category C, the Karoubi envelope construction yields a pseudo-abelian category kar(C) called the pseudo-abelian completion of C. Moreover, the functor :C\rightarrow kar(C) is in fact an additive morphism. To be precise, given a preadditive category C we construct a pseudo-abelian category kar(C) in the following way. The objects of kar(C) are pairs (X,p) where X is an object of C and p is an idempotent of X. The morphisms :f:(X,p)\rightarrow (Y,q) in kar(C) are those morphisms :f:X\rightarrow Y such that f=q\circ f = f \circ p in C. The functor :C\rightarrow kar(C) is given by taking X to (X,id_X).

In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module F1 and a surjective homomorphism p1 : F1 → X. Then one finds a free module F2 and a surjective homomorphism p2 : F2 → ker(p1). Continuing in this fashion, a sequence of free modules Fn and homomorphisms pn can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology Hn of this complex depends only on F and X and is, by definition, the n-th derived functor of F, applied to X. A common use of group (co)homology H^2(G,M)is to classify the possible extension groups E which contain a given G-module M as a normal subgroup and have a given quotient group G, so that G = E/M.

Let C be a category, and let F be a contravariant functor from C to the category of sets Set. A contravariant functor G from C to Set is a subfunctor of F if # For all objects c of C, G(c) ⊆ F(c), and # For all arrows f: c′ → c of C, G(f) is the restriction of F(f) to G(c). This relation is often written as G ⊆ F. For example, let 1 be the category with a single object and a single arrow. A functor F: 1 → Set maps the unique object of 1 to some set S and the unique identity arrow of 1 to the identity function 1S on S. A subfunctor G of F maps the unique object of 1 to a subset T of S and maps the unique identity arrow to the identity function 1T on T. Notice that 1T is the restriction of 1S to T. Consequently, subfunctors of F correspond to subsets of S.

Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K (assumed to be unital and associative). Let :U : K-Alg -> K-Vect be the forgetful functor which assigns to each algebra its underlying vector space. Given any vector space V over K we can construct the tensor algebra T(V). The tensor algebra is characterized by the fact: :“Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T(V) to A.” This statement is an initial property of the tensor algebra since it expresses the fact that the pair (T(V),i), where i:V \to U(T(V)) is the inclusion map, is a universal morphism from the vector space V to the functor U. Since this construction works for any vector space V, we conclude that T is a functor from K-Vect to K-Alg.

This functor takes an object c′ of C and gives back all of the morphisms c′ → c. A subfunctor of gives back only some of the morphisms. Such a subfunctor is called a sieve, and it is usually used when defining Grothendieck topologies.

Free objects are all examples of a left adjoint to a forgetful functor which assigns to an algebraic object its underlying set. These algebraic free functors have generally the same description as in the detailed description of the free group situation above.

Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept.

Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functors preserve all limits. It follows that any functor which fails to preserve some limit is not representable. Contravariant representable functors take colimits to limits.

The Bredon cohomology, introduced by Glen E. Bredon, is a type of equivariant cohomology that is a contravariant functor from the category of G-complex with equivariant homotopy maps to the category of abelian groups together with the connecting homomorphism satisfying some conditions.

The divided power envelope is a fundamental tool in the theory of PD differential operators and crystalline cohomology, where it is used to overcome technical difficulties which arise in positive characteristic. The divided power functor is used in the construction of co-Schur functors.

Functors are very useful in modeling functional effects to apply a function to computations that did not yet finish. Functors form a base for more complex abstractions like Applicative, Monad, Comonad. In C++, the name functor refers to a function object instead of this definition.

She placed great emphasis on the order in which reading symptoms emerged (poor nonword reading first, then visual errors, then noun > functor, then noun > verb, concrete > abstract, and finally, semantic errors) and suggested that the continuum hypothesis was supported by this pattern of symptoms.

In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form.

As is well known, the two alethic functors could be replaced by a single dyadic functor with the following syntax and semantics: if α and β are formulas, then (αβ) is a formula whose semantics are "not (α and/or β)" (see NAND and NOR).

A functor between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams. That is, if is a biproduct of in C with projection morphisms and injection morphisms , then should be a biproduct of in D with projection morphisms and injection morphisms . Almost all functors studied between additive categories are additive. In fact, it is a theorem that all adjoint functors between additive categories must be additive functors (see here), and most interesting functors studied in all of category theory are adjoints.

D-modules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves. For a map f: X → Y of smooth varieties, the definitions are this: :DX->Y := OX ⊗f−1(OY) f−1(DY) This is equipped with a left DX action in a way that emulates the chain rule, and with the natural right action of f−1(DY). The pullback is defined as :f∗(M) := DX->Y ⊗f−1(DY) f−1(M). Here M is a left DY-module, while its pullback is a left module over X. This functor is right exact, its left derived functor is denoted Lf∗.

The representability theorem for CW complexes, due to Edgar H. Brown, is the following. Suppose that: # The functor F maps coproducts (i.e. wedge sums) in Hotc to products in Set: F(\vee_\alpha X_\alpha) \cong \prod_\alpha F(X_\alpha), # The functor F maps homotopy pushouts in Hotc to weak pullbacks. This is often stated as a Mayer–Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements u ∈ F(U), v ∈ F(V) such that u and v restrict to the same element of F(U ∩ V), there is an element w ∈ F(W) restricting to u and v, respectively.

Given a uniformizable space X there is a finest uniformity on X compatible with the topology of X called the fine uniformity or universal uniformity. A uniform space is said to be fine if it has the fine uniformity generated by its uniform topology. The fine uniformity is characterized by the universal property: any continuous function f from a fine space X to a uniform space Y is uniformly continuous. This implies that the functor F : CReg → Uni that assigns to any completely regular space X the fine uniformity on X is left adjoint to the forgetful functor sending a uniform space to its underlying completely regular space.

The functor T assigns to a context G a set of types, and for each , a set of terms. The axioms for a functor require that these play harmoniously with substitution. Substitution is usually written in the form Af or af, where A is a type in and a is a term in , and f is a substitution from D to G. Here and . The category C must contain a terminal object (the empty context), and a final object for a form of product called comprehension, or context extension, in which the right element is a type in the context of the left element.

If V is a vector space over a field F, then the cofree coalgebra C (V), of V, is a coalgebra together with a linear map C (V) → V, such that any linear map from a coalgebra X to V factors through a coalgebra homomorphism from X to C (V). In other words, the functor C is right adjoint to the forgetful functor from coalgebras to vector spaces. The cofree coalgebra of a vector space always exists, and is unique up to canonical isomorphism. Cofree cocommutative coalgebras are defined in a similar way, and can be constructed as the largest cocommutative coalgebra in the cofree coalgebra.

They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets. There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.

If A is an abelian category and A is an object of A, then HomA(A,-) is a covariant left-exact functor from A to the category Ab of abelian groups. It is exact if and only if A is projective.Jacobson (2009), p. 149, Prop. 3.9.

The full subcategory of B-admissible representations, denoted RepB(G), is Tannakian. If B has extra structure, such as a filtration or an E-linear endomorphism, then DB(V) inherits this structure and the functor DB can be viewed as taking values in the corresponding category.

These correspond to the daughter and mother relations. Left nodes are addressed as 0 nodes and right nodes are 1 nodes. By convention, nodes on the left correspond to argument nodes, i.e. nodes in which arguments are represented, whereas right nodes correspond to the functor nodes, i.e.

It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.

Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B. If 0 → A → B → C → 0 is a short exact sequence in A, then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) there is one canonical way of doing so, given by the right derived functors of F. For every i≥1, there is a functor RiF: A → B, and the above sequence continues like so: 0 → F(A) → F(B) → F(C) → R1F(A) → R1F(B) → R1F(C) → R2F(A) → R2F(B) → ... . From this we see that F is an exact functor if and only if R1F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact.

Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F2 of the formal group law. :[x,y] = F2(x,y) − F2(y,x) The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed by taking the Lie algebra of the formal group: ::Lie groups -> Formal group laws -> Lie algebras Over fields of characteristic 0, formal group laws are essentially the same as finite- dimensional Lie algebras: more precisely, the functor from finite-dimensional formal group laws to finite-dimensional Lie algebras is an equivalence of categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact, in this case it is well known that passing from an algebraic group to its Lie algebra often throws away too much information, but passing instead to the formal group law often keeps enough information.

The GSL can be used in C++ classes, but not using pointers to member functions, because the type of pointer to member function is different from pointer to function.Pointers to member functions. . Instead, pointers to static functions have to be used. Another common workaround is using a functor.

The category of fields, denoted Field, is the full subcategory of CRing whose objects are fields. The category of fields is not nearly as well-behaved as other algebraic categories. In particular, free fields do not exist (i.e. there is no left adjoint to the forgetful functor Field → Set).

A map between Lawvere theories (L, I) and (L′, I′) is a finite-product preserving functor that commutes with I and I′. Such a map is commonly seen as an interpretation of (L, I) in (L′, I′). Lawvere theories together with maps between them form the category Law.

Like many categories, the category Manp is a concrete category, meaning its objects are sets with additional structure (i.e. a topology and an equivalence class of atlases of charts defining a Cp-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor :U : Manp -> Top to the category of topological spaces which assigns to each manifold the underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor :U′ : Manp -> Set to the category of sets which assigns to each manifold the underlying set and to each p-times continuously differentiable function the underlying function.

Finally, the entry in the array at the key position is returned to the caller. The above strategy requires explicit wrapping at each call to a function that is to be memoized. In those languages that allow closures, memoization can be effected implicitly by a functor factory that returns a wrapped memoized function object. In pseudocode, this can be expressed as follows: function construct-memoized-functor (F is a function object parameter) allocate a function object called memoized-version; let memoized-version(arguments) be if self has no attached array values then [self is a reference to this object] allocate an associative array called values; attach values to self; end if; if self.

Given a topological space , a local system is a functor from the fundamental groupoid of to a category.Spanier, chapter 1; Exercises F. As an important special case, a bundle of (abelian) groups on is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on assigns a group to each element of , and assigns a group homomorphism to each continuous path from to . In order to be a functor, these group homomorphisms are required to be compatible with the topological structure, so that homotopic paths with fixed endpoints define the same homomorphism; furthermore the group homomorphisms must compose in accordance with the concatenation and inversion of paths.

These terms come from the notion of covariant and contravariant functors in category theory. Consider the category C whose objects are types and whose morphisms represent the subtype relationship ≤. (This is an example of how any partially ordered set can be considered as a category.) Then for example the function type constructor takes two types p and r and creates a new type p → r; so it takes objects in C^2 to objects in C. By the subtyping rule for function types this operation reverses ≤ for the first parameter and preserves it for the second, so it is a contravariant functor in the first parameter and a covariant functor in the second.

In the language of representable functors one can state the above result as follows. The contravariant functor, which associates to each k-variety T the set of families of degree 0 line bundles parametrised by T and to each k-morphism f: T → T' the mapping induced by the pullback with f, is representable. The universal element representing this functor is the pair (Av, P). This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms fv: Bv → Av in a compatible way.

Thus any continuous function f from a space X to a space Y defines an inverse mapping f −1 from Ω(Y) to Ω(X). Furthermore, it is easy to check that f −1 (like any inverse image map) preserves finite intersections and arbitrary unions and therefore is a morphism of frames. If we define Ω(f) = f −1 then Ω becomes a contravariant functor from the category Top to the category Frm of frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor from Frm to Top which is adjoint to Ω.

The fundamental groups of surfaces, as mentioned above, can also be computed using this theorem. In the parlance of category theory, the theorem can be concisely stated by saying that the fundamental group functor takes pushouts (in the category of topological spaces) along inclusions to pushouts (in the category of groups).

In SGA4, tome I, section 1, the notion of left (right) exact functors are defined for general categories, and not just abelian ones. The definition is as follows: :Let C be a category with finite projective (resp. inductive) limits. Then a functor from C to another category C′ is left (resp.

This embedding admits a left adjoint functor F. The images of F and G are isomorphic, an isomorphism being obtained by restricting F and G to those images. The category of d-spaces can thus be seen as one of the most general formalisations of the intuitive notion of directed space.

There are two hypercohomology spectral sequences; one with E2 term :R^iF(H^j(C)) and the other with E1 term :R^jF(C^i) and E2 term :H^i(R^jF(C)) both converging to the hypercohomology :H^{i+j}(RF(C)), where RjF is a right derived functor of F.

Finally, there is (limited, as of April 2020) support for allowing the user to create classes to represent a parametrised function as a functor. While not strictly wrappers, there are some C++ classes o2scl Object-oriented Scientific Computing Library; yat. that allow C++ users to use the Gnu Scientific Library with wrapper features.

In mathematics, a pseudofunctor f is a mapping between categories that is just like a functor except that f(x \circ y) = f(x) \circ f(y) and f(1) = 1 do not hold as exact equalities but only up to coherent isomorphisms. The Grothendieck construction associates to a pseudofunctor a fibered category.

A Frobenioid consists of a category C together with a functor to an elementary Frobenioid, satisfying some complicated conditions related to the behavior of line bundles and divisors on models of global fields. One of Mochizuki's fundamental theorems states that under various conditions a Frobenioid can be reconstructed from the category C.

The dimension group of an AF algebra is a Riesz group. The Effros- Handelman-Shen theorem says the converse is true. Every Riesz group, with a given scale, arises as the dimension group of some AF algebra. This specifies the range of the classifying functor K0 for AF-algebras and completes the classification.

This is the standard concept. Heuristically, if we have a space M for which each point m ∊ M corresponds to an algebro-geometric object Um, then we can assemble these objects into a tautological family U over M. (For example, the Grassmannian G(k, V) carries a rank k bundle whose fiber at any point [L] ∊ G(k, V) is simply the linear subspace L ⊂ V.) M is called a base space of the family U. We say that such a family is universal if any family of algebro-geometric objects T over any base space B is the pullback of U along a unique map B → M. A fine moduli space is a space M which is the base of a universal family. More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of all suitable families of objects with base B. A space M is a fine moduli space for the functor F if M represents F, i.e., there is a natural isomorphism τ : F → Hom(−, M), where Hom(−, M) is the functor of points.

If a complete lattice is freely generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset. The definition of the result of this operation is similar to the above definition of free objects, where "sets" and "functions" are replaced by "posets" and "monotone mappings". Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction. As long as one considers meet- or join-preserving functions as morphisms, this can easily be achieved through the so-called Dedekind–MacNeille completion.

By the above, we can define the category of cones to F as the comma category (Δ ↓ F). Morphisms of cones are then just morphisms in this category. This equivalence is rooted in the observation that a natural map between constant functors Δ(N), Δ(M) corresponds to a morphism between N and M. In this sense, the diagonal functor acts trivially on arrows. In similar vein, writing down the definition of a natural map from a constant functor Δ(N) to F yields the same diagram as the above. As one might expect, a morphism from a cone (N, ψ) to a cone (L, φ) is just a morphism N → L such that all the "obvious" diagrams commute (see the first diagram in the next section).

The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups G which are profinite groups. The gist of the theory, which is rather elaborate in detail in the exposition of Saavedra Rivano, is that the fiber functor Φ of the Galois theory is replaced by a tensor functor T from C to K-Vect. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group (a priori only a monoid) of natural transformations of T into itself, that respect the tensor structure. This is by nature not an algebraic group, but an inverse limit of algebraic groups (pro-algebraic group).

The above examples of localization of R-modules is abstracted in the following definition. In this shape, it applies in many more examples, some of which are sketched below. Given a category C and some class W of morphisms in C, the localization C[W−1] is another category which is obtained by inverting all the morphisms in W. More formally, it is characterized by a universal property: there is a natural localization functor C -> C[W−1] and given another category D, a functor F: C -> D factors uniquely over C[W−1] if and only if F sends all arrows in W to isomorphisms. Thus, the localization of the category is unique up to unique isomorphism of categories, provided that it exists.

If an abelian category has enough injectives, we can form injective resolutions, i.e. for a given object X we can form a long exact sequence :0\to X \to Q^0 \to Q^1 \to Q^2 \to \cdots and one can then define the derived functors of a given functor F by applying F to this sequence and computing the homology of the resulting (not necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology and algebraic geometry. The categories being used are typically functor categories or categories of sheaves of OX modules over some ringed space (X, OX) or, more generally, any Grothendieck category.

The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor :U : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint :F : Set → Ring which assigns to each set X the free ring generated by X. One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are forgetful functors :A : Ring → Ab :M : Ring → Mon which "forget" multiplication and addition, respectively.

Z of the fundamental group of the punctured disk. The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor Φ, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type :G ≅ Aut(Φ), the latter being the group of automorphisms (self-natural equivalences) of Φ. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite. To see how this applies to the case of fields, one has to study the tensor product of fields.

In mathematics, Brown's representability theorem in homotopy theory, see pages 152–157 gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor. More specifically, we are given :F: Hotcop → Set, and there are certain obviously necessary conditions for F to be of type Hom(--, C), with C a pointed connected CW-complex that can be deduced from category theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of pointed sets; in other words the sets are also given a base point.

It is a simple exercise in topology to see that every three elements of a Puppe sequence are, up to a homotopy, of the form: : X\to Y\to C(f). By "up to a homotopy", we mean here that every 3 elements in a Puppe sequence are of the above form if regarded as objects and morphisms in the homotopy category. If one is now given a topological half-exact functor, the above property implies that, after acting with the functor in question on the Puppe sequence associated to A\to B, one obtains a long exact sequence. A result, due to John Milnor,John Milnor "Construction of Universal Bundles I" (1956) Annals of Mathematics, 63 pp. 272-284.

If this condition is satisfied for all elements of the locale, then the locale is spatial, or said to have enough points. (See also well-pointed category for a similar condition in more general categories.) Finally, one can verify that for every space X, Ω(X) is spatial and for every locale L, pt(L) is sober. Hence, it follows that the above adjunction of Top and Loc restricts to an equivalence of the full subcategories Sob of sober spaces and SLoc of spatial locales. This main result is completed by the observation that for the functor pt o Ω, sending each space to the points of its open set lattice is left adjoint to the inclusion functor from Sob to Top.

A DG category C is called pre-triangulated if it has a suspension functor \Sigma and a class of distinguished triangles compatible with the suspension, such that its homotopy category Ho(C) is a triangulated category. A triangulated category T is said to have a dg enhancement C if C is a pretriangulated dg category whose homotopy category is equivalent to T.See for a survey of existence and unicity results of dg enhancements dg enhancements. dg enhancements of an exact functor between triangulated categories are defined similarly. In general, there need not exist dg enhancements of triangulated categories or functors between them, for example stable homotopy category can be shown not to arise from a dg category in this way.

The pre-order can be obtained as the specialization pre-order of the McKinsey-Tarski topology. The Esakia duality can be recovered via a functor that replaces the field of sets with the Boolean space it generates. Via a functor that instead replaces the pre-order with its corresponding Alexandrov topology, an alternative representation of the interior algebra as a field of sets is obtained where the topology is the Alexandrov bico- reflection of the McKinsey-Tarski topology. The approach of formulating a topological duality for interior algebras using both the Stone topology of the Jónsson–Tarski approach and the Alexandrov topology of the pre-order to form a bi-topological space has been investigated by G. Bezhanishvili, R.Mines, and P.J. Morandi.

In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra.

Nearly all triangulated categories that arise in practice come from stable ∞-categories. A similar (but more special) enrichment of triangulated categories is the notion of a dg-category. In some ways, stable ∞-categories or dg-categories work better than triangulated categories. One example is the notion of an exact functor between triangulated categories, discussed below.

The STL includes classes that overload the function call operator (). Instances of such classes are called functors or function objects. Functors allow the behavior of the associated function to be parameterized (e.g. through arguments passed to the functor's constructor) and can be used to keep associated per-functor state information along with the function.

Namely, Cl can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.

For details, see . has also shown how one can couple the d = 3 and d = 1 theories together: this is quite analogous to the coupling between d = 2 and d = 0 in Jones–Witten theory. Now, topological field theory is viewed as a functor, not on a fixed dimension but on all dimensions at the same time.

The coproduct of two commutative rings is given by the tensor product of rings. Again, the coproduct of two nonzero commutative rings can be zero. The opposite category of CRing is equivalent to the category of affine schemes. The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum, an affine scheme.

Any class of objects defines a discrete category when augmented with identity maps. Any subcategory of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full. The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct.

In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.

In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

A diagram is said to be small or finite whenever J is. A morphism of diagrams of type J in a category C is a natural transformation between functors. One can then interpret the category of diagrams of type J in C as the functor category CJ, and a diagram is then an object in this category.

Given a topological group G, i.e., a group equipped with a topology such that product and inverse are continuous maps, it is natural to consider continuous G-modules, i.e., requiring that the action :G \times M \to M is a continuous map. For such modules, one can again consider the derived functor of M \mapsto M^G.

We can fix this by separating it into two diagrams, one in BA and one in Set. To relate the two, we introduce a functor U : BA → Set that "forgets" the algebraic structure, mapping algebras and homomorphisms to their underlying sets and functions. center If we interpret the top arrow as a diagram in BA and the bottom triangle as a diagram in Set, then this diagram properly expresses that every function f : X → UB extends to a unique Boolean algebra homomorphism f′ : FX → B. The functor U can be thought of as a device to pull the homomorphism f′ back into Set so it can be related to f. The remarkable aspect of this is that the latter diagram is one of the various (equivalent) definitions of when two functors are adjoint.

The precise statement is as follows: if A is a small abelian category, then there exists a ring R (with 1, not necessarily commutative) and a full, faithful and exact functor F: A → R-Mod (where the latter denotes the category of all left R-modules). The functor F yields an equivalence between A and a full subcategory of R-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R-Mod. Such an equivalence is necessarily additive. The theorem thus essentially says that the objects of A can be thought of as R-modules, and the morphisms as R-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules.

Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold. From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow its dual .

A cone with vertex N of a diagram D : J → C is a morphism from the constant diagram Δ(N) to D. The constant diagram is the diagram which sends every object of J to an object N of C and every morphism to the identity morphism on N. The limit of a diagram D is a universal cone to D. That is, a cone through which all other cones uniquely factor. If the limit exists in a category C for all diagrams of type J one obtains a functor which sends each diagram to its limit. Dually, the colimit of diagram D is a universal cone from D. If the colimit exists for all diagrams of type J one has a functor which sends each diagram to its colimit.

Diagrams and functor categories are often visualized by commutative diagrams, particularly if the index category is a finite poset category with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be expressed as compositions. The commutativity corresponds to the uniqueness of a map between two objects in a poset category. Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way. Not every diagram commutes, as not every index category is a poset category: most simply, the diagram of a single object with an endomorphism or with two parallel arrows (\bullet \rightrightarrows \bullet; f,g\colon X \to Y) need not commute.

The completion of a signalizer functor has a "good chance" of being normal in G, according to the top of the article. Here, the coprime action fact will be used to motivate this claim. Let \theta be a complete A-signalizer functor on G Let B be a noncyclic subgroup of A. Then the coprime action fact together with the balance condition imply that W= \langle \theta(a) \mid a \in A, a eq 1\rangle = \langle \theta(b) \mid b \in B, b eq 1\rangle . To see this, observe that because \theta(a) is B-invariant, we have \theta(a) = \langle \theta(a) \cap C_G(b) \mid b \in B, b eq 1\rangle \subseteq \langle \theta(b) \mid b \in B, b eq 1\rangle.

A subgroupoid is a subcategory that is itself a groupoid. A groupoid morphism is simply a functor between two (category- theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd. It is useful that this category is, like the category of small categories, Cartesian closed.

Abstracting yet again, some diagrammatic and/or sequential constructions are often "naturally related" – a vague notion, at first sight. This leads to the clarifying concept of natural transformation, a way to "map" one functor to another. Many important constructions in mathematics can be studied in this context. "Naturality" is a principle, like general covariance in physics, that cuts deeper than is initially apparent.

There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives. This is exactly the formula . There are also chain rules in stochastic calculus. One of these, Itō's lemma, expresses the composite of an Itō process (or more generally a semimartingale) dXt with a twice-differentiable function f.

Formally, an absolute coequalizer of a pair of parallel arrows f, g : X → Y in a category C is a coequalizer as defined above, but with the added property that given any functor F: C → D, F(Q) together with F(q) is the coequalizer of F(f) and F(g) in the category D. Split coequalizers are examples of absolute coequalizers.

In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.

A direct system or an ind-object in a category C is defined to be a functor :F : I \to C from a small filtered category I to C. For example, if I is the category N mentioned above, this datum is equivalent to a sequence :X_0 \to X_1 \to \cdots of objects in C together with morphisms as displayed.

Waldhausen introduced the idea of a trace map from the algebraic K-theory of a ring to its Hochschild homology; by way of this map, information can be obtained about the K-theory from the Hochschild homology. Bökstedt factorized this trace map, leading to the idea of a functor known as the topological Hochschild homology of the ring's Eilenberg–MacLane spectrum.

Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi- isomorphism. (The converse is false in general.) This shows that there is a canonical functor K(A) \rightarrow D(A) to the derived category (if A is abelian).

In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers)Johannes Stern, Toward Predicate Approaches to Modality, Springer, 2015, p. 11. that operate on terms to yield terms.

A deeper application coming from Kuiper's theorem is the proof of the Atiyah–Jänich theorem (after Klaus Jänich and Michael Atiyah), stating that the space of Fredholm operators on H, with the norm topology, represents the functor K(.) of topological (complex) K-theory, in the sense of homotopy theory. This is given by Atiyah.Michael Atiyah, K-theory p. 153 and p.

So at each point, an element of a fixed vector space is assigned. However, sheaves can "continuously change" the vector space (or more generally abelian group). This entire process is really the global section functor, which assigns to each sheaf its global section. Then sheaf cohomology enables us to consider a similar extension problem while "continuously varying" the abelian group.

Then F and G can be restricted to D1 and C1 and yield inverse equivalences of these subcategories. In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of F (i.e. a functor G such that FG is naturally isomorphic to 1D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.

The exterior derivative is natural in the technical sense: if is a smooth map and is the contravariant smooth functor that assigns to each manifold the space of -forms on the manifold, then the following diagram commutes :none so , where denotes the pullback of . This follows from that , by definition, is , being the pushforward of . Thus is a natural transformation from to .

The 0-graded component of the free Lie algebra is just the free vector space on that set. One can alternatively define a free Lie algebra on a vector space V as left adjoint to the forgetful functor from Lie algebras over a field K to vector spaces over the field K – forgetting the Lie algebra structure, but remembering the vector space structure.

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

A monad is a certain type of endofunctor. For example, if F and G are a pair of adjoint functors, with F left adjoint to G, then the composition G \circ F is a monad. If F and G are inverse functors, the corresponding monad is the identity functor. In general, adjunctions are not equivalences--they relate categories of different natures.

In mathematics, an extensive category is a category C with finite coproducts that are disjoint and well-behaved with respect to pullbacks. Equivalently, C is extensive if the coproduct functor from the product of the slice categories C/X × C/Y to the slice category C/(X + Y) is an equivalence of categories for all objects X and Y of C.

In mathematics, an exp algebra is a Hopf algebra Exp(G) constructed from an abelian group G, and is the universal ring R such that there is an exponential map from G to the group of the power series in Rt with constant term 1\. In other words the functor Exp from abelian groups to commutative rings is adjoint to the functor from commutative rings to abelian groups taking a ring to the group of formal power series with constant term 1\. The definition of the exp ring of G is similar to that of the group ring Z[G] of G, which is the universal ring such that there is an exponential homomorphism from the group to its units. In particular there is a natural homomorphism from the group ring to a completion of the exp ring.

Fix an arbitrary field k and let Fields/k denote the category of finitely generated field extensions of k with inclusions as morphisms. Consider a (covariant) functor F : Fields/k → Set. For a field extension K/k and an element a of F(K/k) a field of definition of a is an intermediate field K/L/k such that a is contained in the image of the map F(L/k) → F(K/k) induced by the inclusion of L in K. The essential dimension of a, denoted by ed(a), is the least transcendence degree (over k) of a field of definition for a. The essential dimension of the functor F, denoted by ed(F), is the supremum of ed(a) taken over all elements a of F(K/k) and objects K/k of Fields/k.

Then F is representable by some CW complex C, that is to say there is an isomorphism :F(Z) ≅ HomHotc(Z, C) for any CW complex Z, which is natural in Z in that for any morphism from Z to another CW complex Y the induced maps F(Y) → F(Z) and HomHot(Y, C) → HomHot(Z, C) are compatible with these isomorphisms. The converse statement also holds: any functor represented by a CW complex satisfies the above two properties. This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication. The representing object C above can be shown to depend functorially on F: any natural transformation from F to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects.

A point of a topos X is defined as a geometric morphism from the topos of sets to X. If X is an ordinary space and x is a point of X, then the functor that takes a sheaf F to its stalk Fx has a right adjoint (the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map x: 1 → X. More precisely, those are the global points. They are not adequate in themselves for displaying the space-like aspect of a topos, because a non- trivial topos may fail to have any. Generalized points are geometric morphisms from a topos Y (the stage of definition) to X. There are enough of these to display the space-like aspect.

Let C and D be categories. The collection of all functors from C to D forms the objects of a category: the functor category. Morphisms in this category are natural transformations between functors. Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.

Let C be a category and let J be a Grothendieck topology on C. The pair (C, J) is called a site. A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology.

A category is itself a type of mathematical structure, so we can look for "processes" which preserve this structure in some sense; such a process is called a functor. Diagram chasing is a visual method of arguing with abstract "arrows" joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions. Functors can define (construct) categorical diagrams and sequences (viz.

In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids.

Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function between vector spaces is entirely determined by its values on a basis of the vector space E1. The following definition translates this to any category. A concrete category is a category that is equipped with a faithful functor to Set, the category of sets.

For example, the quanta of the electromagnetic field are identified with photons. Unlike first quantization, conventional second quantization is completely unambiguous, in effect a functor. Historically, quantizing the classical theory of a single particle gave rise to a wavefunction. The classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave-function of one of its quanta.

Suppose C is any category and A, T are two objects of C. A T-valued point of A is simply an arrow p \colon T \to A. The set of all T-valued points of A varies functorially with T, giving rise to the "functor of points" of A; according to the Yoneda lemma, this completely determines A as an object of C.

In the mid-1970s, Enright introduced new methods that led him to an algebraic way of looking at discrete series (which were fundamental representations constructed by Harish-Chandra in the early 1960s), and to an algebraic proof of the Blattner multiplicity formula. He was known for Enright–Varadarajan modules, Enright resolutions, and the Enright completion functor, which has had a lasting influence in algebra.

There is a wide generalization of the Cauchy–Kovalevskaya theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kovalevskaya–Kashiwara theorem, due to . This theorem involves a cohomological formulation, presented in the language of D-modules. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a derived functor Ext^1.

The functor from Top to CGTop that takes X to Xc is right adjoint to the inclusion functor CGTop → Top. The continuity of a map defined on a compactly generated space X can be determined solely by looking at the compact subsets of X. Specifically, a function f : X → Y is continuous if and only if it is continuous when restricted to each compact subset K ⊆ X. If X and Y are two compactly generated spaces the product X × Y may not be compactly generated (it will be if at least one of the factors is locally compact). Therefore when working in categories of compactly generated spaces it is necessary to define the product as (X × Y)c. The exponential object in CGHaus is given by (YX)c where YX is the space of continuous maps from X to Y with the compact-open topology.

If C is an additive category and we require the congruence relation ~ on C to be additive (i.e. if f1, f2, g1 and g2 are morphisms from X to Y with f1 ~ f2 and g1 ~g2, then f1 \+ f2 ~ g1 \+ g2), then the quotient category C/~ will also be additive, and the quotient functor C -> C/~ will be an additive functor. The concept of an additive congruence relation is equivalent to the concept of a two-sided ideal of morphisms: for any two objects X and Y we are given an additive subgroup I(X,Y) of HomC(X, Y) such that for all f ∈ I(X,Y), g ∈ HomC(Y, Z) and h∈ HomC(W, X), we have gf ∈ I(X,Z) and fh ∈ I(W,Y). Two morphisms in HomC(X, Y) are congruent iff their difference is in I(X,Y).

In mathematics, specifically category theory, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

These sheaves admit algebraic operations which are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory which can remember higher information, in the same way that derived functors in homological algebra yield higher information about operations such as tensor product and the Hom functor on modules.

For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension. Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism T \to S of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.

Suppose that is the structure morphism for an -scheme . The base scheme has a Frobenius morphism FS. Composing with FS results in an -scheme XF called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an -morphism induces an -morphism . For example, consider a ring A of characteristic and a finitely presented algebra over A: :R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m).

From a category-theoretic point of view, the fundamental group is a functor :{Pointed algebraic varieties} -> {Profinite groups}. The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions). Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups. studies higher étale homotopy groups by means of the étale homotopy type of a scheme.

Bundles may also be described by their sheaves of sections. The pullback of bundles then corresponds to the inverse image of sheaves, which is a contravariant functor. A sheaf, however, is more naturally a covariant object, since it has a pushforward, called the direct image of a sheaf. The tension and interplay between bundles and sheaves, or inverse and direct image, can be advantageous in many areas of geometry.

Using natural transformations as morphisms, one can form the category of all representations of G in C. This is just the functor category CG. For another example, take C = Top, the category of topological spaces. A representation of G in Top is a topological space on which G acts continuously. An equivariant map is then a continuous map f : X → Y between representations which commutes with the action of G.

For instance, we say that the categories C and D are equivalent (respectively dually equivalent) if there exists an equivalence (respectively duality) between them. Furthermore, we say that F "is" an equivalence of categories if an inverse functor G and natural isomorphisms as above exist. Note however that knowledge of F is usually not enough to reconstruct G and the natural isomorphisms: there may be many choices (see example below).

Like any other discipline, mathematics also has its own brand of technical terminology. In some cases, a word in general usage can have a different and specific meaning within mathematics (such as the cases of "group", "ring", "field", "category", "term" and "factor"). For more examples, see :Category:Mathematical terminology. In other cases, specialist terms, such as "tensor", "fractal" and "functor", have been created exclusively for the use in mathematics.

The quiver itself can be considered a category, where the vertices are objects and paths are morphisms. Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces. Morphisms of representations of Q are precisely natural transformations between the corresponding functors. For a finite quiver Γ (a quiver with finitely many vertices and edges), let KΓ be its path algebra.

Using the language of category theory, the composition operator is a pull-back on the space of measurable functions; it is adjoint to the transfer operator in the same way that the pull-back is adjoint to the push-forward; the composition operator is the inverse image functor. Since the domain considered here is that of Borel functions, the above describes the Koopman operator as it appears in Borel functional calculus.

More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, the set X(S) of S-points of X means the set of morphisms Spec(S) → X over Spec(R). The scheme X is determined up to isomorphism by the functor S ↦ X(S); this is the philosophy of identifying a scheme with its functor of points. Another formulation is that the scheme X over R determines a scheme XS over S by base change, and the S-points of X (over R) can be identified with the S-points of XS (over S). The theory of Diophantine equations traditionally meant the study of integral points, meaning solutions of polynomial equations in the integers Z rather than the rationals Q. For homogeneous polynomial equations such as x3 \+ y3 = z3, the two problems are essentially equivalent, since every rational point can be scaled to become an integral point.

For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to X. There is a rather minor set-theoretical problem here, since Et(X) is a "large" category: its objects do not form a set. However, it is equivalent to a small category because étale morphisms are locally of finite presentation, so it is harmless to pretend that it is a small category. A presheaf on a topological space X is a contravariant functor from the category of open subsets to sets. By analogy we define an étale presheaf on a scheme X to be a contravariant functor from Et(X) to sets.

The natural transformations from V' to an arbitrary graph (functor) G constitute the vertices of G while those from E' to G constitute its edges. Although SetC, which we can identify with Grph, is not made concrete by either V' or E' alone, the functor U: Grph -> Set2 sending object G to the pair of sets (Grph(V' ,G), Grph(E' ,G)) and morphism h: G -> H to the pair of functions (Grph(V' ,h), Grph(E' ,h)) is faithful. That is, a morphism of graphs can be understood as a pair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of generalized elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of topoi by allowing an object to have multiple underlying sets, that is, to be multisorted.

A flat resolution of a module M is a resolution of the form :\cdots \to F_2 \to F_1 \to F_0 \to M \to 0, where the Fi are all flat modules. Any free or projective resolution is necessarily a flat resolution. Flat resolutions can be used to compute the Tor functor. The length of a finite flat resolution is the first subscript n such that Fn is nonzero and Fi = 0 for i > n.

In differential algebra, the derivative is interpreted as a morphism of modules of Kähler differentials. A ring homomorphism of commutative rings determines a morphism of Kähler differentials which sends an element dr to d(f(r)), the exterior differential of f(r). The formula holds in this context as well. The common feature of these examples is that they are expressions of the idea that the derivative is part of a functor.

C++ templates are completely type safe at compile time. As a demonstration, the standard type `complex` does not define the `<` operator, because there is no strict order on complex numbers. Therefore, `max(x, y)` will fail with a compile error, if x and y are `complex` values. Likewise, other templates that rely on `<` cannot be applied to `complex` data unless a comparison (in the form of a functor or function) is provided. E.g.

The notion of a sheaf and sheafification of a presheaf date to early category theory, and can be seen as the linear form of the calculus of functors. The quadratic form can be seen in the work of André Haefliger on links of spheres in 1965, where he defined a "metastable range" in which the problem is simpler. This was identified as the quadratic approximation to the embeddings functor in Goodwillie and Weiss.

One also talks about injective objects in categories more general than module categories, for instance in functor categories or in categories of sheaves of OX-modules over some ringed space (X,OX). The following general definition is used: an object Q of the category C is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g.

Fulton and MacPherson extended the Chow ring to singular varieties by defining the "operational Chow ring" and more generally a bivariant theory associated to any morphism of schemes.Fulton, Intersection Theory, Chapter 17. A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively. It generalizes a cohomology theory, which is a contravariant functor that assigns to a space a ring, namely a cohomology ring.

Let D be a triangulated category with arbitrary direct sums. A localizing subcategory of D is a strictly full triangulated subcategory that is closed under arbitrary direct sums.Neeman (2001), Introduction, after Remark 1.4. To explain the name: if a localizing subcategory S of a compactly generated triangulated category D is generated by a set of objects, then there is a Bousfield localization functor L\colon D\to D with kernel S.Krause (2010), Theorem, Introduction.

The product does not necessarily exist. For example, an empty product (i.e. is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group there are infinitely many morphisms , so cannot be terminal. If is a set such that all products for families indexed with exist, then one can treat each product as a functor .

In category theory, there are covariant functors and contravariant functors. The assignment of the dual space to a vector space is a standard example of a contravariant functor. Some constructions of multilinear algebra are of "mixed" variance, which prevents them from being functors. In differential geometry, the components of a vector relative to a basis of the tangent bundle are covariant if they change with the same linear transformation as a change of basis.

For Hironaka's variety V over the complex numbers with an automorphism of order 2 as above, the Hilbert functor HilbV/C of closed subschemes is not representable by a scheme, essentially because the quotient by the group of order 2 does not exist as a scheme . In other words, this gives an example of a smooth complete variety whose Hilbert scheme does not exist. Grothendieck showed that the Hilbert scheme always exists for projective varieties.

Another important property of Galois connections is that lower adjoints preserve all suprema that exist within their domain. Dually, upper adjoints preserve all existing infima. From these properties, one can also conclude monotonicity of the adjoints immediately. The adjoint functor theorem for order theory states that the converse implication is also valid in certain cases: especially, any mapping between complete lattices that preserves all suprema is the lower adjoint of a Galois connection.

In many programming languages, map is the name of a higher-order function that applies a given function to each element of a functor, e.g. a list, returning a list of results in the same order. It is often called apply-to-all when considered in functional form. The concept of a map is not limited to lists: it works for sequential containers, tree-like containers, or even abstract containers such as futures and promises.

The companion concept to associated bundles is the reduction of the structure group of a G-bundle B. We ask whether there is an H-bundle C, such that the associated G-bundle is B, up to isomorphism. More concretely, this asks whether the transition data for B can consistently be written with values in H. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).

In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category. In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.

For a separated presheaf, the first arrow need only be injective. Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.

Mitchell, 1965). A functor associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second. As a result, this defines a category of categories and functors – the objects are categories, and the morphisms (between categories) are functors. Studying categories and functors is not just studying a class of mathematical structures and the morphisms between them but rather the relationships between various classes of mathematical structures.

Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in A are made equivalent to 0, when considered as chains in X. Heuristically, one often thinks of a pair (X,A) as being akin to the quotient space X/A. There is a functor from spaces to pairs, which sends a space X to the pair (X,\varnothing). A related concept is that of a triple , with . Triples are used in homotopy theory.

Graphs with labels attached to edges or vertices are more generally designated as labeled. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled. (In the literature, the term labeled may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.) The category of all graphs is the slice category Set ↓ D where D: Set → Set is the functor taking a set s to s × s.

Let R be a ring. Write R-Mod for the category of left R-modules and Mod-R for the category of right R-modules. (If R is commutative, the two categories can be identified.) For a fixed left R-module B, let T(A) = A ⊗R B for A in Mod-R. This is a right exact functor from Mod-R to the category of abelian groups Ab, and so it has left derived functors LiT.

Cartan and Eilenberg showed that these constructions are independent of the choice of projective resolution, and that both constructions yield the same Tor groups.Weibel (1994), section 2.4 and Theorem 2.7.2. Moreover, for a fixed ring R, Tor is a functor in each variable (from R-modules to abelian groups). For a commutative ring R and R-modules A and B, Tor(A, B) is an R-module (using that A ⊗R B is an R-module in this case).

In mathematics, a preradical is a subfunctor of the identity functor in the category of left modules over a ring with identity. The class of all preradicals over R-mod is denoted by R-pr. There is a natural order in R-pr given by, for any two preradicals \sigma and \tau , \sigma\leq\tau, if for any left R-module M, \sigma M\leq \tau M. With this order R-pr becomes a big lattice.

Many trim functions have an optional parameter to specify a list of characters to trim, instead of the default whitespace characters. For example, PHP and Python allow this optional parameter, while Pascal and Java do not. With Common Lisp's `string-trim` function, the parameter (called character-bag) is required. The C++ Boost library defines space characters according to locale, as well as offering variants with a predicate parameter (a functor) to select which characters are trimmed.

Work of Grothendieck and David Mumford (see geometric invariant theory) opened up this area in the early 1960s. The more algebraic and abstract approach to moduli problems is to set them up as a representable functor question, then apply a criterion that singles out the representable functors for schemes. When this programmatic approach works, the result is a fine moduli scheme. Under the influence of more geometric ideas, it suffices to find a scheme that gives the correct geometric points.

An interesting situation occurs if a function preserves all suprema (or infima). More accurately, this is expressed by saying that a function preserves all existing suprema (or infima), and it may well be that the posets under consideration are not complete lattices. For example, (monotone) Galois connections have this property. Conversely, by the order theoretical Adjoint Functor Theorem, mappings that preserve all suprema/infima can be guaranteed to be part of a unique Galois connection as long as some additional requirements are met.

In functional programming, an anamorphism is a generalization of the concept of unfolds on coinductive lists. Formally, anamorphisms are generic functions that can corecursively construct a result of a certain type and which is parameterized by functions that determine the next single step of the construction. The data type in question is defined as the greatest fixed point ν X . F X of a functor F. By the universal property of final coalgebras, there is a unique coalgebra morphism A → ν X .

The mod p cohomology of the classifying spaces of certain finite groups (elementary Abelian p-groups, for which the generalized Sullivan conjecture was formulated) played an important role in the proof. The connection between the cohomology theory of these finite groups and the classifying spaces of groups is illuminated by the work of Lannes. He introduced the T-functor on the category of unstable algebra over the Steenrod algebra. Lannes thus led an important development of algebraic topology in the 1980s.

Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves. An alternative approach to the dual point of view is to use the functor of points.

So one can talk about a sheaf of sets on a site, a sheaf of abelian groups on a site, and so on. The definition of sheaf cohomology as a derived functor also works on a site. So one has sheaf cohomology groups Hj(X, E) for any object X of a site and any sheaf E of abelian groups. For the étale topology, this gives the notion of étale cohomology, which led to the proof of the Weil conjectures.

The Alexandroff extension can be viewed as a functor from the category of topological spaces with proper continuous maps as morphisms to the category whose objects are continuous maps c\colon X \rightarrow Y and for which the morphisms from c_1\colon X_1 \rightarrow Y_1 to c_2\colon X_2 \rightarrow Y_2 are pairs of continuous maps f_X\colon X_1 \rightarrow X_2, \ f_Y\colon Y_1 \rightarrow Y_2 such that f_Y \circ c_1 = c_2 \circ f_X. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

Similar ideas occur in category theory: the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category. A three-dimensional Euclidean space is a special case of a Euclidean space. In Bourbaki's terms, the species of three- dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space. Fig.

In algebra, an action of a monoidal category S on a category X is a functor :\cdot: S \times X \to X such that there are natural isomorphisms s \cdot (t \cdot x) \simeq (s \cdot t)\cdot x and e \cdot x \simeq x and those natural isomorphism satisfy the coherence conditions analogous to those in S. If there is such an action, S is said to act on X. For example, S acts on itself via the monoid operation ⊗.

The use of templates allowed for compile time typesafe verification of connections. The addition of this strict compile time checking required the addition of template typecasting adapters which convert the functor callback profile to match the required signal pattern. libsigc++ was a natural expansion of the C++ standard library functors to the tracking of objects necessary to implement the observer pattern. It inspired multiple C++ template based signal and slot implementations including the signal implementation used in the boost C++ libraries.

In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data. Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric interpretation.

A span is a diagram of type \Lambda = (-1 \leftarrow 0 \rightarrow +1), i.e., a diagram of the form Y \leftarrow X \rightarrow Z. That is, let Λ be the category (-1 ← 0 → +1). Then a span in a category C is a functor S : Λ → C. This means that a span consists of three objects X, Y and Z of C and morphisms f : X → Y and g : X → Z: it is two maps with common domain. The colimit of a span is a pushout.

In Itō's lemma, the derivative of the composite function depends not only on dXt and the derivative of f but also on the second derivative of f. The dependence on the second derivative is a consequence of the non-zero quadratic variation of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way. This variant of the chain rule is not an example of a functor because the two functions being composed are of different types.

Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form ΠE. The word "noncanonically" prevents one from concluding that supermanifolds are simply glorified vector bundles; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an equivalence of categories. It was published by Marjorie Batchelor in 1979. The proof of Batchelor's theorem relies in an essential way on the existence of a partition of unity, so it does not hold for complex or real-analytic supermanifolds.

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.

In category theory, a branch of mathematics, a stable model category is a pointed model category in which the suspension functor is an equivalence of the homotopy category with itself. The prototypical examples are the category of spectra in the stable homotopy theory and the category of chain complex of R-modules. On the other hand, the category of pointed topological spaces and the category of pointed simplicial sets are not stable model categories. Any stable model category is equivalent to a category of presheaves of spectra.

The pointwise sum of two rng homomorphisms is generally not a rng homomorphism. There is a fully faithful functor from the category of abelian groups to Rng sending an abelian group to the associated rng of square zero. Free constructions are less natural in Rng than they are in Ring. For example, the free rng generated by a set {x} is the ring of all integral polynomials over x with no constant term, while the free ring generated by {x} is just the polynomial ring Z[x].

The current domain of a variable can be inspected using specific literals; for example, `dom(X,D)` finds out the current domain `D` of a variable `X`. As for domains of reals, functors can be used with domains of integers. In this case, a term can be an expression over integers, a constant, or the application of a functor over other terms. A variable can take an arbitrary term as a value, if its domain has not been specified to be a set of integers or constants.

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology.

If is a homomorphism of rings of characteristic , then :\phi(x^p) = \phi(x)^p. If and are the Frobenius endomorphisms of and , then this can be rewritten as: :\phi \circ F_R = F_S \circ \phi. This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic rings to itself. If the ring is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: means , which by definition means that is nilpotent of order at most .

An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoretic point of view, a monoid is a category with one object, and an act is a functor from that category to the category of sets. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets. Another important special case is a transformation semigroup.

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.

It is possible to use the tensor algebra to describe the symmetric algebra . In fact, can be defined as the quotient algebra of by the two sided ideal generated by the commutators v\otimes w - w\otimes v. It is straightforward, but rather tedious, to verify that the resulting algebra satisfies the universal property stated in the introduction. This results also directly from a general result of category theory, which asserts that the composition of two left adjoint functors is also a left adjoint functor.

In algebraic geometry, the Serre–Tate theorem, says that an abelian scheme and its p-divisible group have the same infinitesimal deformation theory. This was first proved by Serre when the reduction of the abelian variety is ordinary, using the Greenberg functor; then Tate gave a proof in the general case by a different method. Their proofs were not published, but they were summarized in the notes of the Lubin–Serre–Tate seminar (Woods Hole, 1964). Other proofs were published by Messing (1962) and Drinfeld (1976).

The basic types in Fril are similar to those in Prolog, with one important exception: Prolog's compound data type is the term, with lists defined as nested terms using the `.` functor; in Fril, the compound type is the list itself, which forms the basis for most constructs. Variables are distinguished by identifiers containing only uppercase letters and underscores (whereas Prolog only requires the first character to be uppercase). As in Prolog, the name `_` is reserved to mean "any value", with multiple occurrences of `_` replaced by distinct variables.

Given an object X, a functor G (taking for simplicity the first functor to be the identity) and an isomorphism \eta\colon X \to G(X), proof of unnaturality is most easily shown by giving an automorphism A\colon X \to X that does not commute with this isomorphism (so \eta \circ A eq G(A) \circ \eta). More strongly, if one wishes to prove that X and G(X) are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism \eta, there is some A with which it does not commute; in some cases a single automorphism A works for all candidate isomorphisms \eta while in other cases one must show how to construct a different A_\eta for each isomorphism. The maps of the category play a crucial role – any infranatural transform is natural if the only maps are the identity map, for instance. This is similar (but more categorical) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is "not natural", or rather "not unique", as automorphisms exist that do not preserve the direct sum decomposition – see for example.

The original construction of tmf uses the obstruction theory of Hopkins, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf Otop ("top" stands for topological) of multiplicative cohomology theories on the etale site of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology theory) whose associated formal group is the formal group of that elliptic curve. A second construction, due to Jacob Lurie, constructs tmf rather by describing the moduli problem it represents and applying general representability theory to then show existence: just as the moduli stack of elliptic curves represents the functor that assigns to a ring the category of elliptic curves over it, the stack together with the sheaf of E-infinity ring spectra represents the functor that assigns to an E-infinity ring its category of oriented derived elliptic curves, appropriately interpreted.

More generally, bundles or bundle objects can be defined in any category: in a category C, a bundle is simply an epimorphism π: E → B. If the category is not concrete, then the notion of a preimage of the map is not necessarily available. Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do; for instance, for a category with pullbacks and a terminal object 1 the points of B can be identified with morphisms p:1→B and the fiber of p is obtained as the pullback of p and π. The category of bundles over B is a subcategory of the slice category (C↓B) of objects over B, while the category of bundles without fixed base object is a subcategory of the comma category (C↓C) which is also the functor category C², the category of morphisms in C. The category of smooth vector bundles is a bundle object over the category of smooth manifolds in Cat, the category of small categories. The functor taking each manifold to its tangent bundle is an example of a section of this bundle object.

The notion is an analog of a classifying space in algebraic topology. In algebraic topology, the basic fact is that each principal G-bundle over a space S is (up to natural isomorphisms) the pullback of a universal bundle EG \to BG along some map from S to BG. In other words, to give a principal G-bundle over a space S is the same as to give a map (called a classifying map) from a space S to the classifying space BG of G. A similar phenomenon in algebraic geometry is given by a linear system: to give a morphism from a projective variety to a projective space is (up to base loci) to give a linear system on the projective variety. Yoneda's lemma says that a scheme X determines and is determined by its points.In fact, X is determined by its R-points with various rings R: in the precise terms, given schemes X, Y, any natural transformation from the functor R \mapsto X(R) to the functor R \mapsto Y(R) determines a morphism of schemes X →Y in a natural way.

In mathematics, the category of magmas, denoted Mag, is the category whose objects are magmas (that is, sets equipped with a binary operation), and whose morphisms are magma homomorphisms. The category Mag has direct products, so the concept of a magma object (internal binary operation) makes sense. (As in any category with direct products.) There is an inclusion functor: as trivial magmas, with operations given by projection: . An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.

In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D (the identity functor on D) and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one-to-one correspondence to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.

In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. It is named after Peter Goddard and Charles Thorn. The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the natural inner product induced on the output vector space is positive definite. Thus, there were no so-called ghosts (Pauli–Villars ghosts), or vectors of negative norm.

There are two naturally isomorphic functors that are typically used to quantize bosonic strings. In both cases, one starts with positive-energy representations of the Virasoro algebra of central charge 26, equipped with Virasoro-invariant bilinear forms, and ends up with vector spaces equipped with bilinear forms. Here, "Virasoro-invariant" means Ln is adjoint to L−n for all integers n. The first functor historically is "old canonical quantization", and it is given by taking the quotient of the weight 1 primary subspace by the radical of the bilinear form.

Here, "primary subspace" is the set of vectors annihilated by Ln for all strictly positive n, and "weight 1" means L0 acts by identity. A second, naturally isomorphic functor, is given by degree 1 BRST cohomology. Older treatments of BRST cohomology often have a shift in the degree due to a change in choice of BRST charge, so one may see degree −1/2 cohomology in papers and texts from before 1995. A proof that the functors are naturally isomorphic can be found in Section 4.4 of Polchinski's String Theory text.

Conversely, any invariant subset of X is a union of orbits. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit. A G-invariant element of X is such that for all . The set of all such x is denoted XG and called the G-invariants of X. When X is a G-module, XG is the zeroth cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants.

If one starts with a covariant right-exact functor G, and the category A has enough projectives (i.e. for every object A of A there exists an epimorphism P → A where P is a projective object), then one can define analogously the left-derived functors LiG. For an object X of A we first construct a projective resolution of the form :\cdots\to P_2\to P_1\to P_0 \to X \to 0 where the Pi are projective. We apply G to this sequence, chop off the last term, and compute homology to get LiG(X).

As before, L0G(X) = G(X). In this case, the long exact sequence will grow "to the left" rather than to the right: :0\to A \to B \to C \to 0 is turned into :\cdots\to L_2G(C) \to L_1G(A) \to L_1G(B)\to L_1G(C)\to G(A)\to G(B)\to G(C)\to 0. Left derived functors are zero on all projective objects. One may also start with a contravariant left-exact functor F; the resulting right-derived functors are then also contravariant.

The explicitly foundational role of the category Set in his treatment is noteworthy in view of the folk intuition that enriched categories liberate category theory from the last vestiges of Set as the codomain of the ordinary external hom-functor. In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales. In 1972 he was elected a Fellow of the Australian Academy of Science. He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1994.

This duality means that in addition to the correspondence between Boolean algebras and their Stone spaces, each homomorphism from a Boolean algebra A to a Boolean algebra B corresponds in a natural way to a continuous function from S(B) to S(A). In other words, there is a contravariant functor that gives an equivalence between the categories. This was an early example of a nontrivial duality of categories. The theorem is a special case of Stone duality, a more general framework for dualities between topological spaces and partially ordered sets.

In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object. Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks.

The third fact above required the development of new techniques in motivic homotopy theory. The goal was to prove that a functor, which was not assumed to commute with limits or colimits, preserved weak equivalences between objects of a certain form. One of the main difficulties there was that the standard approach to the study of weak equivalences is based on Bousfield–Quillen factorization systems and model category structures, and these were inadequate. Other methods had to be developed, and this work was completed by Voevodsky only in 2008.

Functors, or function objects, are similar to function pointers, and can be used in similar ways. A functor is an object of a class type that implements the function-call operator, allowing the object to be used within expressions using the same syntax as a function call. Functors are more powerful than simple function pointers, being able to contain their own data values, and allowing the programmer to emulate closures. They are also used as callback functions if it is necessary to use a member function as a callback function.

Every module M also has an injective resolution: an exact sequence of the form :0 → M → I0 → I1 → I2 → ... where the I j are injective modules. Injective resolutions can be used to define derived functors such as the Ext functor. The length of a finite injective resolution is the first index n such that In is nonzero and Ii = 0 for i greater than n. If a module M admits a finite injective resolution, the minimal length among all finite injective resolutions of M is called its injective dimension and denoted id(M).

Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.

The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not exist any faithful functor from hTop to Set was first proven by Peter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalence-classes of functors" also fails to be concretizable.

Given a concrete category (C, U) and a cardinal number N, let UN be the functor C → Set determined by UN(c) = (U(c))N. Then a subfunctor of UN is called an N-ary predicate and a natural transformation UN → U an N-ary operation. The class of all N-ary predicates and N-ary operations of a concrete category (C,U), with N ranging over the class of all cardinal numbers, forms a large signature. The category of models for this signature then contains a full subcategory which is equivalent to C.

In some parts of category theory, most notably topos theory, it is common to replace the category Set with a different category X, often called a base category. For this reason, it makes sense to call a pair (C, U) where C is a category and U a faithful functor C → X a concrete category over X. For example, it may be useful to think of the models of a theory with N sorts as forming a concrete category over SetN. In this context, a concrete category over Set is sometimes called a construct.

These values are programmable, however, and some default values that can be used to enforce the systematicity principle are described in [Falkenhainer et al., 1989]. These rules are: #If the source and target are not functions and have the same order, the match gets +0.3 evidence. If the orders are within 1 of each other, the match gets +0.2 evidence and -0.05 evidence. #If the source and target have the same functor, the match gets 0.2 evidence if the source is a function and 0.5 if the source is a relation.

In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.

It is also possible to start with the functor F, and pose the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. This gives the intuition behind the fact that adjoint functors occur in pairs: if F is left adjoint to G, then G is right adjoint to F.

The tension between these two motivations was especially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, who used category theory to take compass bearings in other work—in functional analysis, homological algebra and finally algebraic geometry. It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role of adjunction was inherent in Grothendieck's approach. For example, one of his major achievements was the formulation of Serre duality in relative form—loosely, in a continuous family of algebraic varieties.

If a functor F : D → C is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms. Every adjunction 〈F, G, ε, η〉 extends an equivalence of certain subcategories. Define C1 as the full subcategory of C consisting of those objects X of C for which εX is an isomorphism, and define D1 as the full subcategory of D consisting of those objects Y of D for which ηY is an isomorphism.

The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms.) This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes.Haantjes, J., & Laman, G. (1953). On the definition of geometric objects. I. Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles...

Defining a quantity does not guarantee its existence. Given a functor F: C \to D and an object X of C, there may or may not exist a universal morphism from X to F. If, however, a universal morphism (A, u) does exist, then it is essentially unique. Specifically, it is unique up to a unique isomorphism: if (A', u') is another pair, then there exists a unique isomorphism k: A \to A' such that u' = F(k) \circ u. This is easily seen by substituting (A, u') in the definition of a universal morphism.

It is then straightforward to show that contains and satisfies the above universal property. As a consequence of this construction, the operation of assigning to a vector space its exterior algebra is a functor from the category of vector spaces to the category of algebras. Rather than defining first and then identifying the exterior powers as certain subspaces, one may alternatively define the spaces first and then combine them to form the algebra . This approach is often used in differential geometry and is described in the next section.

The finite-dimensional representations of an algebraic group G, together with the tensor product of representations, form a tannakian category RepG. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine group scheme over a field k is pro-algebraic in the sense that it is an inverse limit of affine group schemes of finite type over k.Deligne & Milne (1982), Corollary II.2.7.) For example, the Mumford–Tate group and the motivic Galois group are constructed using this formalism.

In representation theory, the category of representations of some algebraic structure has the representations of as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an object decomposes into simple objects (see Maschke's theorem for the case of finite groups). The Tannakian formalism gives conditions under which a group G may be recovered from the category of representations of it together with the forgetful functor to the category of vector spaces.

In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first-order logic) by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory of (the codomain of) a forgetful functor, and in physics of (hypothesized) hidden variable theories purporting to explain quantum mechanics. Elementary classes are (vacuously) pseudoelementary but the converse is not always true; nevertheless pseudoelementary classes share some of the properties of elementary classes such as being closed under ultraproducts.

Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite category. A diagram is said to be small or finite whenever J is.

Let A be a Grothendieck category (an AB5 category with a generator), G a generator of A and R be the ring of endomorphisms of G; also, let S be the functor from A to Mod-R (the category of right R-modules) defined by S(X) = Hom(G,X). Then the Gabriel–Popescu theorem states that S is full and faithful and has an exact left adjoint. This implies that A is equivalent to the Serre quotient category of Mod-R by a certain localizing subcategory C. (A localizing subcategory of Mod-R is a full subcategory C of Mod-R, closed under arbitrary direct sums, such that for any short exact sequence of modules 0\rarr M_1\rarr M_2\rarr M_3\rarr 0, we have M2 in C if and only if M1 and M3 are in C. The Serre quotient of Mod-R by any localizing subcategory is a Grothendieck category.) We may take C to be the kernel of the left adjoint of the functor S. Note that the embedding S of A into Mod-R is left-exact but not necessarily right-exact: cokernels of morphisms in A do not in general correspond to the cokernels of the corresponding morphisms in Mod-R.

Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top. Top is also fiber-complete meaning that the category of all topologies on a given set X (called the fiber of U above X) forms a complete lattice when ordered by inclusion. The greatest element in this fiber is the discrete topology on X, while the least element is the indiscrete topology.

He used étale coverings to define an algebraic analogue of the fundamental group of a topological space. Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions that imitated the cohomology functor H1. Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory that he suspected would be the Weil cohomology. To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead.

The above property characterizes free groups up to isomorphism, and is sometimes used as an alternative definition. It is known as the universal property of free groups, and the generating set S is called a basis for FS. The basis for a free group is not uniquely determined. Being characterized by a universal property is the standard feature of free objects in universal algebra. In the language of category theory, the construction of the free group (similar to most constructions of free objects) is a functor from the category of sets to the category of groups.

The syntactic type of a lexical item can be either a primitive type, such as S, N, or NP, or complex, such as S\NP, or NP/N. The complex types, schematizable as X/Y and X\Y, denote functor types that take an argument of type Y and return an object of type X. A forward slash denotes that the argument should appear to the right, while a backslash denotes that the argument should appear on the left. Any type can stand in for the X and Y here, making syntactic types in CCG a recursive type system.

A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be built up (or faithfully represented up to homotopy) from simplices and their incidence relations. This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology. To get back to actual topological spaces, there is a geometric realization functor which turns simplicial sets into compactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory are generalized by analogous results for simplicial sets.

Let BordM be the category whose morphisms are n-dimensional submanifolds of M and whose objects are connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are homotopic via submanifolds of M, and so form the quotient category hBordM: The objects in hBordM are the objects of BordM, and the morphisms of hBordM are homotopy equivalence classes of morphisms in BordM. A TQFT on M is a symmetric monoidal functor from hBordM to the category of vector spaces. Note that cobordisms can, if their boundaries match, be sewn together to form a new bordism.

The constant presheaf with value Z, which we will denote F, is the presheaf that chooses all four sets to be Z, the integers, and all restriction maps to be the identity. F is a functor, hence a presheaf, because it is constant. F satisfies the gluing axiom, but it is not a sheaf because it fails the local identity axiom on the empty set. This is because the empty set is covered by the empty family of sets: Vacuously, any two sections of F over the empty set are equal when restricted to any set in the empty family.

This makes rings F-algebras on the category of sets with signature 1 + 1 + R + R×R + R×R. Alternatively, we can look at the functor F(R) = 1 + R×R in the category of abelian groups. In that context, the multiplication is a homomorphism, meaning m(x + y, z) = m(x,z) + m(y,z) and m(x,y + z) = m(x,y) + m(x,z), which are precisely the distributivity conditions. Therefore, a ring is an F-algebra of signature 1 + R×R over the category of abelian groups which satisfies two axioms (associativity and identity for the multiplication).

When we come to vector spaces and modules, the signature functor includes a scalar multiplication k×E -> E, and the signature F(E) = 1 + E + k×E is parametrized by k over the category of fields, or rings. Algebras over a field can be viewed as F-algebras of signature 1 + 1 + A + A×A + A×A + k×A over the category of sets, of signature 1 + A×A over the category of modules (a module with an internal multiplication), and of signature k×A over the category of rings (a ring with a scalar multiplication), when they are associative and unitary.

Then Ext(A, B) is the cohomology of this complex at position i. Cartan and Eilenberg showed that these constructions are independent of the choice of projective or injective resolution, and that both constructions yield the same Ext groups.Weibel (1994), sections 2.4 and 2.5 and Theorem 2.7.6. Moreover, for a fixed ring R, Ext is a functor in each variable (contravariant in A, covariant in B). For a commutative ring R and R-modules A and B, Ext(A, B) is an R-module (using that HomR(A, B) is an R-module in this case).

Equality constraints on terms can be simplified, that is solved, via unification: A constraint `t1=t2` can be simplified if both terms are function symbols applied to other terms. If the two function symbols are the same and the number of subterms is also the same, this constraint can be replaced with the pairwise equality of subterms. If the terms are composed of different function symbols or the same functor but on different number of terms, the constraint is unsatisfiable. If one of the two terms is a variable, the only allowed value the variable can take is the other term.

More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory. For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ2/2. (This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom.

With a monad, a programmer can turn a complicated sequence of functions into a succinct pipeline that abstracts away auxiliary data management, control flow, or side-effects. Both the concept of a monad and the term originally come from category theory, where it is defined as a functor with additional structure. Research beginning in the late 1980s and early 1990s established that monads could bring seemingly disparate computer-science problems under a unified, functional model. Category theory also provides a few formal requirements, known as the monad laws, which should be satisfied by any monad and can be used to verify monadic code.

In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration).Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag (See Chapter 11 for construction.) Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.

Let f\colon A \to B be a continuous map between CW complexes and let C(f) denote a mapping cone of f, (i.e., the cofiber of the map f), so that we have a (cofiber) sequence: :A\to B\to C(f). Now we can form \Sigma A and \Sigma B, suspensions of A and B respectively, and also \Sigma f \colon \Sigma A \to \Sigma B (this is because suspension might be seen as a functor), obtaining a sequence: :\Sigma A \to \Sigma B \to C(\Sigma f). Note that suspension preserves cofiber sequences.

As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : C → D from every object of D, then G has a left adjoint. However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D (equivalently, every object of C).

Given \theta, certain additional, relatively mild, assumptions allow one to prove that the subgroup W= \langle \theta(a) \mid a \in A, a eq 1\rangle of G generated by the subgroups \theta(a) is in fact a p'-subgroup. The Solvable Signalizer Functor Theorem proved by Glauberman and mentioned above says that this will be the case if \theta is solvable and A has at least three generators. The theorem also states that under these assumptions, W itself will be solvable. Several earlier versions of the theorem were proven: proved this under the stronger assumption that A had rank at least 5.

Then the groups Hi(X,E) for integers i are defined as the right derived functors of the functor E ↦ E(X). This makes it automatic that Hi(X,E) is zero for i < 0, and that H0(X,E) is the group E(X) of global sections. The long exact sequence above is also straightforward from this definition. The definition of derived functors uses that the category of sheaves of abelian groups on any topological space X has enough injectives; that is, for every sheaf E there is an injective sheaf I with an injection E → I.Iversen (1986), Theorem II.3.1.

Since both functors and function pointers can be invoked using the syntax of a function call, they are interchangeable as arguments to templates when the corresponding parameter only appears in function call contexts. A particularly common type of functor is the predicate. For example, algorithms like take a unary predicate that operates on the elements of a sequence. Algorithms like sort, partial_sort, nth_element and all sorted containers use a binary predicate that must provide a strict weak ordering, that is, it must behave like a membership test on a transitive, non reflexive and asymmetric binary relation.

In computer science, Cayley representations can be applied to improve the asymptotic efficiency of semigroups by reassociating multiple composed multiplications. The action given by left multiplication results in right-associated multiplication, and vice versa for the action given by right multiplication. Despite having the same results for any semigroup, the asymptotic efficiency will differ. Two examples of useful transformation monoids given by an action of left multiplication are the functional variation of the difference list data structure, and the monadic Codensity transformation (a Cayley representation of a monad, which is a monoid in a particular monoidal functor category).

Algebraic spaces over the complex numbers are closely related to analytic spaces and Moishezon manifolds. Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the étale topology, while analytic spaces are formed by gluing with the classical topology. In particular there is a functor from complex algebraic spaces of finite type to analytic spaces. Hopf manifolds give examples of analytic surfaces that do not come from a proper algebraic space (though one can construct non-proper and non-separated algebraic spaces whose analytic space is the Hopf surface).

In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its integral homology groups: : completely determine its homology groups with coefficients in , for any abelian group : : Here might be the simplicial homology, or more generally the singular homology: the result itself is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients may be used, at the cost of using a Tor functor. For example it is common to take to be , so that coefficients are modulo 2.

Let F : J → C be a diagram in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of objects and morphisms in C. The category J is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when J is a discrete category, it corresponds most closely to the idea of an indexed family in set theory.

As a consequence of the Mayer-Vietoris sequence, the value of an excisive functor on a space X only depends on its value on 'small' subspaces of X, together with the knowledge how these small subspaces intersect. In a cycle representation of the associated homology theory, this means that all cycles must be representable by small cycles. For instance, for singular homology, the excision property is proved by subdivision of simplices, obtaining sums of small simplices representing arbitrary homology classes. In this spirit, for certain homotopy-invariant functors which are not excisive, the corresponding excisive theory may be constructed by imposing 'control conditions', leading to the field of controlled topology.

In mathematics, the category of medial magmas, also known as the medial category, and denoted Med, is the category whose objects are medial magmas (that is, sets with a medial binary operation), and whose morphisms are magma homomorphisms (which are equivalent to homomorphisms in the sense of universal algebra). The category Med has direct products, so the concept of a medial magma object (internal binary operation) makes sense. As a result, Med has all its objects as medial objects, and this characterizes it. There is an inclusion functor from Set to Med as trivial magmas, with operations being the right projections : (x, y) -> y.

An abelian étale sheaf F on X is called finite locally constant if it is a representable functor which can be represented by an étale cover of X. It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant. It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves. Grothendieck originally introduced the machinery of Grothendieck topologies and topoi to define the étale topology.

Here is an analogy: with the Taylor series method from calculus, you can approximate the shape of a smooth function f around a point x by using a sequence of increasingly accurate polynomial functions. In a similar way, with the calculus of functors method, you can approximate the behavior of certain kind of functor F at a particular object X by using a sequence of increasingly accurate polynomial functors. To be specific, let M be a smooth manifold and let O(M) be the category of open subspaces of M—i.e. the category where the objects are the open subspaces of M, and the morphisms are inclusion maps.

The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V: Wn → Wn+1, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected p-group schemes correspond to D-modules for which F is nilpotent, and étale group schemes correspond to modules for which F is an isomorphism.

In the mathematical theory of categories, a sketch is a category D, together with a set of cones intended to be limits and a set of cocones intended to be colimits. A model of the sketch in a category C is a functor :M:D\rightarrow C that takes each specified cone to a limit cone in C and each specified cocone to a colimit cocone in C. Morphisms of models are natural transformations. Sketches are a general way of specifying structures on the objects of a category, forming a category-theoretic analog to the logical concept of a theory and its models. They allow multisorted models and models in any category.

Strictly speaking, we have defined a right closed monoidal category, since we required that right tensoring with any object A has a right adjoint. In a left closed monoidal category, we instead demand that the functor of left tensoring with any object A :B\mapsto A\otimes B have a right adjoint :B\mapsto(B\Leftarrow A) A biclosed monoidal category is a monoidal category that is both left and right closed. A symmetric monoidal category is left closed if and only if it is right closed. Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is left or right closed.

In fact, the same is true more generally for braided monoidal categories: since the braiding makes A \otimes B naturally isomorphic to B \otimes A, the distinction between tensoring on the left and tensoring on the right becomes immaterial, so every right closed braided monoidal category becomes left closed in a canonical way, and vice versa. We have described closed monoidal categories as monoidal categories with an extra property. One can equivalently define a closed monoidal category to be a closed category with an extra property. Namely, we can demand the existence of a tensor product that is left adjoint to the internal Hom functor.

In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y). Most natural functors, including basic ones like pushforwards and pullbacks, are of this type. These kinds of functors were introduced by in order to prove an equivalence between the derived categories of coherent sheaves on an abelian variety and its dual. That equivalence is analogous to the classical Fourier transform that gives an isomorphism between tempered distributions on a finite-dimensional real vector space and its dual.

Any two isomorphic rings are Morita equivalent. The ring of n-by-n matrices with elements in R, denoted Mn(R), is Morita-equivalent to R for any n > 0. Notice that this generalizes the classification of simple artinian rings given by Artin–Wedderburn theory. To see the equivalence, notice that if X is a left R-module then Xn is an Mn(R)-module where the module structure is given by matrix multiplication on the left of column vectors from X. This allows the definition of a functor from the category of left R-modules to the category of left Mn(R)-modules.

In mathematics, the tensor algebra of a vector space V, denoted T(V) or T(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.

One can, in fact, define the tensor algebra T(V) as the unique algebra satisfying this property (specifically, it is unique up to a unique isomorphism), but one must still prove that an object satisfying this property exists. The above universal property shows that the construction of the tensor algebra is functorial in nature. That is, T is a functor from K-Vect, the category of vector spaces over K, to K-Alg, the category of K-algebras. The functoriality of T means that any linear map between K-vector spaces U and W extends uniquely to a K-algebra homomorphism from T(U) to T(W).

The zeroth algebraic K group K_0(R) of a (not necessarily commutative) ring R is the Grothendieck group of the monoid consisting of isomorphism classes of finitely generated projective modules over R, with the monoid operation given by the direct sum. Then K_0 is a covariant functor from rings to abelian groups. The two previous examples are related: consider the case where R = C^\infty(M) is the ring of complex-valued smooth functions on a compact manifold M. In this case the projective R-modules are dual to vector bundles over M (by the Serre-Swan theorem). Thus K_0(R) and K_0(M) are the same group.

By treating fields of sets on pre-orders as a category in its own right this deep connection can be formulated as a category theoretic duality that generalizes Stone representation without topology. R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and modal frames. Naturman showed that in the case of interior algebras this duality applies to more general topomorphisms and can be factored via a category theoretic functor through the duality with topological fields of sets. The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics.

Regardless of the kind of directed space on considers (pospaces, local pospaces, d-spaces or streams) there is an obvious forgetful functor to the category of topological spaces. Given two directed paths γ and δ, a directed homotopy from γ to δ is a morphism of directed spaces h whose underlying map U(h) is a homotopy –in the usual sense– between the underlying path (topology) U(γ) and U(δ). In algebraic topology, there is a homotopy from α to β if and only if there is a homotopy from β to α. Due to non-reversibility, this is no longer true for directed homotopies.

Every Grothendieck topos is an elementary topos, but the converse is not true (since every Grothendieck topos is cocomplete, which is not required from an elementary topos). The categories of finite sets, of finite G-sets (actions of a group G on a finite set), and of finite graphs are elementary topoi that are not Grothendieck topoi. If C is a small category, then the functor category SetC (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos.

The Goddard–Thorn theorem amounts to the assertion that this quantization functor more or less cancels the addition of two free bosons, as conjectured by Lovelace in 1971. Lovelace's precise claim was that at critical dimension 26, Virasoro-type Ward identities cancel two full sets of oscillators. Mathematically, this is the following claim: Let V be a unitarizable Virasoro representation of central charge 24 with Virasoro-invariant bilinear form, and let π1,1λ be the irreducible module of the R1,1 Heisenberg Lie algebra attached to a nonzero vector λ in R1,1. Then the image of V ⊗ π1,1λ under quantization is canonically isomorphic to the subspace of V on which L0 acts by 1-(λ,λ).

Let C be any category. The Yoneda embedding gives a functor Hom(−, X) for each object X of C. The canonical topology is the biggest (finest) topology such that every representable presheaf, i.e. presheaf of the form Hom(−, X), is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic because it consists of the legs of a colimit cone (under the full diagram on the domains of its constituent morphisms) and these colimits are stable under pullbacks along morphisms in C. A topology that is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical.

For example, this allows us to define F(Zp) with values in the p-adic numbers. The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element g is called group-like if Δg = g ⊗ g and εg = 1, and the group-like elements form a group under multiplication. In the case of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form :D(0) + D(1)x + D(2)x2 + ... for nilpotent elements x.

Given a partially ordered set (S,≤), we can define a simplicial set NS, the nerve of S, as follows: for every object [n] of Δ we set NS([n]) = hompo-set( [n] , S), the order-preserving maps from [n] to S. Every morphism φ:[n]->[m] in Δ is an order preserving map, and via composition induces a map NS(φ) : NS([m]) -> NS([n]). It is straightforward to check that NS is a contravariant functor from Δ to Set: a simplicial set. Concretely, the n-simplices of the nerve NS, i.e. the elements of NSn=NS([n]), can be thought of as ordered length-(n+1) sequences of elements from S: (a0 ≤ a1 ≤ ... ≤ an).

There are three important themes in the categorical approach to logic: ;Categorical semantics: Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions. This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient. R.A.G. Seely's modeling of various impredicative theories, such as system F is an example of the usefulness of categorical semantics. :It was found that the connectives of pre-categorical logic were more clearly understood using the concept of adjoint functor, and that the quantifiers were also best understood using adjoint functors.

In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite order theorem in which a function or a functor, defined in a very general way, can in fact be shown to be a polynomial because of some extraneous condition or symmetry imposed upon it. This article treats two forms of the Peetre theorem. The first is the original version which, although quite useful in its own right, is actually too general for most applications.

To understand this characterization as a universal property, take the category C to be the product category D \times D and define the diagonal functor : \Delta: C \to C \times C by \Delta(X) = (X, X) and \Delta(f: X \to Y) = (f, f). Then (X \times Y, (\pi_1, \pi_2)) is a universal morphism from \Delta to the object (X, Y) of D \times D: if (f, g) is any morphism from (Z, Z) to (X, Y), then it must equal a morphism \Delta(h: Z \to X \times Y) = (h,h) from \Delta(Z) = (Z, Z) to \Delta(X \times Y) = (X \times Y, X \times Y) followed by (\pi_1, \pi_2).

It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product.

The first needs of sheaf theory were for sheaves of abelian groups; so taking the category C as the category of abelian groups was only natural. In applications to geometry, for example complex manifolds and algebraic geometry, the idea of a sheaf of local rings is central. This, however, is not quite the same thing; one speaks instead of a locally ringed space, because it is not true, except in trite cases, that such a sheaf is a functor into a category of local rings. It is the stalks of the sheaf that are local rings, not the collections of sections (which are rings, but in general are not close to being local).

The first was to do with its points: back in the days of projective geometry it was clear that the absence of 'enough' points on an algebraic variety was a barrier to having a good geometric theory (in which it was somewhat like a compact manifold). There was also the difficulty, that was clear as soon as topology took form in the first half of the twentieth century, that the topology of algebraic varieties had 'too few' open sets. The question of points was close to resolution by 1950; Alexander Grothendieck took a sweeping step (invoking the Yoneda lemma) that disposed of it-- naturally at a cost, that every variety or more general scheme should become a functor. It wasn't possible to add open sets, though.

It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B. If 0 → A → B → C → 0 is a short exact sequence in A, then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right.

For any scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X. This is the analog of the category of open subsets of X (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of X. The intersection of two objects corresponds to their fiber product over X. Ét(X) is a large category, meaning that its objects do not form a set. An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. A presheaf F is called an étale sheaf if it satisfies the analog of the usual gluing condition for sheaves on topological spaces.

In functional programming, an applicative functor is a structure intermediate between functors and monads, in that they allow sequencing of functorial computations (unlike plain functors) but without deciding on which computation to perform on the basis of the result of a previous computation (unlike monads). Applicative functors are the programming equivalent of lax monoidal functors with tensorial strength in category theory. Applicative functors were introduced in 2007 by Conor McBride and Ross Paterson in their paper Functional Pearl: applicative programming with effects. Applicative functors first appeared as a library feature in Haskell, but have since spread to other languages as well, including Idris, Agda, OCaml, Scala and F#. Both Glasgow Haskell and Idris now offer language features designed to ease programming with applicative functors.

One can also compose two cobordisms when the end of the first is equal to the start of the second. A n-dimensional topological quantum field theory (TQFT) is a monoidal functor from the category of n-cobordisms to the category of complex vector space (where multiplication is given by the tensor product). In particular, cobordisms between 1-dimensional manifolds (which are unions of circles) are compact surfaces whose boundary has been separated into two disjoint unions of circles. Two-dimensional TQFTs correspond to Frobenius algebras, where the circle (the only connected closed 1-manifold) maps to the underlying vector space of the algebra, while the pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative.

Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.. A further generalization is the idea of a stack. Crudely speaking, algebraic stacks generalize algebraic spaces by having an algebraic group attached to each point, which is viewed as the automorphism group of that point. For example, any action of an algebraic group G on an algebraic variety X determines a quotient stack [X/G], which remembers the stabilizer subgroups for the action of G. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.

The ideas were developed in the period 1955–1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of abstract category theory the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem. The difficulties of algebraic geometry with passage to the quotient are acute. The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence (see FGA) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular.

In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms (such as monomorphism or product) given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory, in particular the use of the Yoneda lemma in this way, is due to Grothendieck, and is often called the method of the functor of points.

For the theory associated to the infinite unitary group, U, the space BU is the classifying space for stable complex vector bundles (a Grassmannian in infinite dimensions). One formulation of Bott periodicity describes the twofold loop space, Ω2BU of BU. Here, Ω is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially BU again; more precisely, :\Omega^2BU\simeq \Z\times BU is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU. An equivalent formulation is :\Omega^2U\simeq U . Either of these has the immediate effect of showing why (complex) topological K-theory is a 2-fold periodic theory.

To understand the definition of a universal construction, it is important to look at examples. Universal constructions were not defined out of thin air, but were rather defined after mathematicians began noticing a pattern in many mathematical constructions (see Examples below). Hence, the definition may not make sense to one at first, but will become clear when one reconciles it with concrete examples. Let F: C \to D be a functor between categories C and D. In what follows, let X be an object of D and A an object of C. A universal morphism from X to F is a unique pair (A, u: X \to F(A)) in D which has the following property, commonly referred to as a universal property.

The category Top of topological spaces and their continuous functions embeds in Chu(Set, 2) in the sense that there exists a full and faithful functor F : Top → Chu(Set, 2) providing for each topological space (X, T) its representation F((X, T)) = (X, ∈, T) as noted above. This representation is moreover a realization in the sense of Pultr and Trnková (1980), namely that the representing Chu space has the same set of points as the represented topological space and transforms in the same way via the same functions. Chu spaces are remarkable for the wide variety of familiar structures they realize. Lafont and Streicher (1991) point out that Chu spaces over 2 realize both topological spaces and coherent spaces (introduced by J.-Y.

In affine geometry, one uses Playfair's axiom to find the line through C1 and parallel to B1B2, and to find the line through B2 and parallel to B1C1: their intersection C2 is the result of the indicated translation. In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting"See also forgetful functor.) the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair's axiom (given a line L and a point P not on L, there is exactly one line parallel to L that passes through P) is fundamental in affine geometry.

We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the same: one converts from one to the other by dualizing, i.e. by changing the direction of all arrows, replacing injective objects with projective ones, and so on. Suppose that A is an abelian category with enough injectives and F a left exact functor to another abelian category B. If C is a complex of objects of A bounded on the left, the hypercohomology :Hi(C) of C (for an integer i) is calculated as follows: # Take a quasi-isomorphism Φ : C → I, here I is a complex of injective elements of A. # The hypercohomology Hi(C) of C is then the cohomology Hi(F(I)) of the complex F(I).

There is another covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined). If A is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup T and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups). In any decomposition of A as a direct sum of a torsion subgroup S and a torsion-free subgroup, S must equal T (but the torsion-free subgroup is not uniquely determined). This is a key step in the classification of finitely generated abelian groups.

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F(S) whose underlying set is Nn where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of Nn; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms. This makes F into a functor from commutative R-algebras S to groups. We can extend the definition of F(S) to some topological R-algebras. In particular, if S is an inverse limit of discrete R algebras, we can define F(S) to be the inverse limit of the corresponding groups.

The first definition of the cyclic homology of a ring A over a field of characteristic zero, denoted :HCn(A) or Hnλ(A), proceeded by the means of an explicit chain complex related to the Hochschild homology complex of A. Connes later found a more categorical approach to cyclic homology using a notion of cyclic object in an abelian category, which is analogous to the notion of simplicial object. In this way, cyclic homology (and cohomology) may be interpreted as a derived functor, which can be explicitly computed by the means of the (b, B)-bicomplex. One of the striking features of cyclic homology is the existence of a long exact sequence connecting Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity sequence.

It is dual to the mapping cone in the sense that the product above is essentially the fibered product or pullback X\times_f Y which is dual to the pushout X\sqcup_f Y used to construct the mapping cone. In this particular case, the duality is essentially that of currying, in that the mapping cone (X\times I)\sqcup_f Y has the curried form X \times_f (I\to Y) where I\to Y is simply an alternate notation for the space Y^I of all continuous maps from the unit interval to Y. The two variants are related by an adjoint functor. Observe that the currying preserves the reduced nature of the maps: in the one case, to the tip of the cone, and in the other case, paths to the basepoint.

The set of morphisms (order-preserving functions) between two preorders actually has more structure than that of a set. It can be made into a preordered set itself by the pointwise relation: : (f ≤ g) ⇔ (∀x f(x) ≤ g(x)) This preordered set can in turn be considered as a category, which makes Ord a 2-category (the additional axioms of a 2-category trivially hold because any equation of parallel morphisms is true in a posetal category). With this 2-category structure, a pseudofunctor F from a category C to Ord is given by the same data as a 2-functor, but has the relaxed properties: : ∀x ∈ F(A), F(idA)(x) ≃ x, : ∀x ∈ F(A), F(g∘f)(x) ≃ F(g)(F(f)(x)), where x ≃ y means x ≤ y and y ≤ x.

The species E• × E• can be seen as making two independent selections from the base set. The two points might coincide, unlike in X·X·E, where they are forced to be different. As functors, species F and G may be combined by functorial composition: (F \,\Box\, G) [A] = F[G[A] ] (the box symbol is used, because the circle is already in use for substitution). This constructs an F-structure on the set of all G-structures on the set A. For example, if F is the functor taking a set to its power set, a structure of the composed species is some subset of the G-structures on A. If we now take G to be E• × E• from above, we obtain the species of directed graphs, with self-loops permitted.

Using the language of category theory, R. A. G. Seely introduced the notion of a locally cartesian closed category (LCCC) as the basic model of type theory. This has been refined by Hofmann and Dybjer to Categories with Families or Categories with Attributes based on earlier work by Cartmell. A category with families is a category C of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor T : Cop → Fam(Set). Fam(Set) is the category of families of Sets, in which objects are pairs of an "index set" A and a function B: X → A, and morphisms are pairs of functions f : A → A' and g : X → X' , such that B' ° g = f ° B in other words, f maps Ba to Bg(a).

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.

Let S and T be simplicial complexes. A simplicial map f from S to T is a function from the vertex set of S to the vertex set of T such that the image of each simplex in S (viewed as a set of vertices) is a simplex in T. A simplicial map f: S → T determines a homomorphism of homology groups Hk(S) → Hk(T) for each integer k. This is the homomorphism associated to a chain map from the chain complex of S to the chain complex of T. Explicitly, this chain map is given on k-chains by :f((v_0, \ldots, v_k)) = (f(v_0),\ldots,f(v_k)) if f(v0), ..., f(vk) are all distinct, and otherwise f((v0, ..., vk)) = 0. This construction makes simplicial homology a functor from simplicial complexes to abelian groups.

This equation says that a tree consists of a single root and a set of (sub-)trees. The recursion does not need an explicit base case: it only generates trees in the context of being applied to some finite set. One way to think about this is that the Ar functor is being applied repeatedly to a "supply" of elements from the set -- each time, one element is taken by X, and the others distributed by E among the Ar subtrees, until there are no more elements to give to E. This shows that algebraic descriptions of species are quite different from type specifications in programming languages like Haskell. Likewise, the species P can be characterised as P = E(E+): "a partition is a pairwise disjoint set of nonempty sets (using up all the elements of the input set)".

The fiber of the corresponding vector bundle over x is then the range of f(x). If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if M is a zero-dimensional 2-point manifold, the module \R\oplus 0 is finitely-generated and projective over C^\infty(M)\cong\R\times\R but is not free, and so cannot correspond to the sections of any (constant-rank) vector bundle over M (all of which are trivial). Another way of stating the above is that for any connected smooth manifold M, the section functor Γ from the category of smooth vector bundles over M to the category of finitely generated, projective C∞(M)-modules is full, faithful, and essentially surjective.

The theory of adjoints has the terms left and right at its foundation, and there are many components which live in one of two categories C and D which are under consideration. Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category C or the "righthand" category D, and also to write them down in this order whenever possible. In this article for example, the letters X, F, f, ε will consistently denote things which live in the category C, the letters Y, G, g, η will consistently denote things which live in the category D, and whenever possible such things will be referred to in order from left to right (a functor F : D → C can be thought of as "living" where its outputs are, in C).

If Y is a group-like H-space, then a product [A, Y] × [B, Y] → [A ∧ B, Y] is defined in analogy with the generalised Whitehead product. This is the generalized Samelson product denoted <σ, τ> for σ ∈ [A, Y] and τ ∈ [B, Y] . If λU,V : [U, ΩV] → [ΣU, V] is the adjoint isomorphism, where Ω is the loop space functor, then λA∧B,X<σ, τ>= [λA,X (σ), λB,X (τ)] for Y = ΩX. An Eckmann–Hilton dual of the generalised Whitehead product can be defined as follows. Let A♭B be the homotopy fiber of the inclusion j : A ∨ B → A × B, that is, the space of paths in A × B which begin in A ∨ B and end at the base point and let γ ∈ [X, ΩA] and δ ∈ [X, ΩB].

A graph consists of two sets, an edge set and a vertex set, and two functions s,t between those sets, assigning to every edge e its source s(e) and target t(e). Grph is thus equivalent to the functor category SetC, where C is the category with two objects E and V and two morphisms s,t: E -> V giving respectively the source and target of each edge. The Yoneda Lemma asserts that Cop embeds in SetC as a full subcategory. In the graph example the embedding represents Cop as the subcategory of SetC whose two objects are V' as the one-vertex no-edge graph and E' as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms from V' to E' (both as natural transformations).

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A). An abelian group A is called a torsion (or periodic) group if every element of A has finite order and is called torsion-free if every element of A except the identity is of infinite order. The proof that AT is closed under the group operation relies on the commutativity of the operation (see examples section). If A is abelian, then the torsion subgroup T is a fully characteristic subgroup of A and the factor group A/T is torsion- free. There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup.

Going one step ahead of universal algebra, most algebraic structures are F-algebras. For example, abelian groups are F-algebras for the same functor F(G) = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t(x,y) = (y,x) is the transpose on GxG. Monoids are F-algebras of signature F(M) = 1 + M×M. In the same vein, semigroups are F-algebras of signature F(S) = S×S Rings, domains and fields are also F-algebras with a signature involving two laws +,•: R×R -> R, an additive identity 0: 1 -> R, a multiplicative identity 1: 1 -> R, and an additive inverse for each element -: R -> R. As all these functions share the same codomain R they can be glued into a single signature function 1 + 1 + R + R×R + R×R -> R, with axioms to express associativity, distributivity, and so on.

Fine moduli spaces are desirable, but they do not always exist and are frequently difficult to construct, so mathematicians sometimes use a weaker notion, the idea of a coarse moduli space. A space M is a coarse moduli space for the functor F if there exists a natural transformation τ : F → Hom(−, M) and τ is universal among such natural transformations. More concretely, M is a coarse moduli space for F if any family T over a base B gives rise to a map φT : B → M and any two objects V and W (regarded as families over a point) correspond to the same point of M if and only if V and W are isomorphic. Thus, M is a space which has a point for every object that could appear in a family, and whose geometry reflects the ways objects can vary in families.

The Grothendieck group ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over semisimple rings, this is no loss as every module is a semisimple module and so a direct sum of simple modules. Ordinary character theory provides better arithmetic control, and uses simple CG modules to understand the structure of finite groups G. Modular representation theory uses Brauer characters to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the Ext functor and describing the module category in various ways including quivers (whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and Auslander–Reiten theory where the associated graph has a vertex for every indecomposable module.

In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external space. The prototypical example of an ∞-topos is the ∞-category of sheaves of spaces on some topological space. But the notion is more flexible; for example, the ∞-category of étale sheaves on some scheme is not the ∞-category of sheaves on any topological space but it is still an ∞-topos. Precisely, in Lurie's Higher Topos Theory, an ∞-topos is defined as an ∞-category X such that there is a small ∞-category C and a left exact localization functor from the ∞-category of presheaves of spaces on C to X. A theorem of Lurie states that an ∞-category is an ∞-topos if and only if it satisfies an ∞-categorical version of Giraud’s axioms in ordinary topos theory.

Let R be a fixed commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is an additive abelian group A which has the structure of both a ring and an R-module in such a way that the scalar multiplication satisfies :r\cdot(xy) = (r\cdot x)y = x(r\cdot y) for all r ∈ R and x, y ∈ A. Furthermore, A is assumed to be unital, which is to say it contains an element 1 such that :1 x = x = x 1 for all x ∈ A. Note that such an element 1 is necessarily unique. In other words, A is an R-module together with a R-bilinear binary operation A × A → A that is associative, and has an identity. Technical note: the multiplicative identity is a datum (there is the forgetful functor from the category of unital associative algebras to the category of possibly non-unital associative algebras) while associativity is a property.

If X is a Klein surface, a function f:X→Cu{∞} is called meromorphic if, on each coordinate patch, f or its complex conjugate is meromorphic in the ordinary sense, and if f takes only real values (or ∞) on the boundary of X. Given a connected Klein surface X, the set of meromorphic functions defined on X form a field M(X), an algebraic function field in one variable over R. M is a contravariant functor and yields a duality (contravariant equivalence) between the category of compact connected Klein surfaces (with non-constant morphisms) and the category of function fields in one variable over the reals. One can classify the compact connected Klein surfaces X up to homeomorphism (not up to equivalence!) by specifying three numbers (g, k, a): the genus g of the analytic double Σ, the number k of connected components of the boundary of X , and the number a, defined by a=0 if X is orientable and a=1 otherwise. We always have k ≤ g+1. The Euler characteristic of X equals 1-g.

However, a procedure to apply any simple function over the whole list, in other words , is straight- forward: (map φ) xlist = [ φ(x1), φ(x2), ..., φ(xn) ] Now, these two procedures already promote `List` to an applicative functor. To fully qualify as a monad, only a correct notion of to flatten repeated structure is needed, but for lists, that just means unwrapping an outer list to append the inner ones that contain values: join(xlistlist) = join([xlist1, xlist2, ..., xlistn]) = xlist1 ++ xlist2 ++ ... ++ xlistn The resulting monad is not only a list, but one that automatically resizes and condenses itself as functions are applied. can now also be derived with just a formula, then used to feed `List` values through a pipeline of monadic functions: (xlist >>= f) = join ∘ (map f) xlist One application for this monadic list is representing nondeterministic computation. `List` can hold results for all execution paths in an algorithm, then condense itself at each step to "forget" which paths led to which results (a sometimes important distinction from deterministic, exhaustive algorithms).

This idea is made formal in the idea of the slice category of objects of C 'above' S. To move from one slice to another requires a base change; from a technical point of view base change becomes a major issue for the whole approach (see for example Beck–Chevalley conditions). A base change 'along' a given morphism :g: T -> S is typically given by the fiber product, producing an object over T from one over S. The 'fiber' terminology is significant: the underlying heuristic is that X over S is a family of fibers, one for each 'point' of S; the fiber product is then the family on T, which described by fibers is for each point of T the fiber at its image in S. This set-theoretic language is too naïve to fit the required context, certainly, from algebraic geometry. It combines, though, with the use of the Yoneda lemma to replace the 'point' idea with that of treating an object, such as S, as 'as good as' the representable functor it sets up. The Grothendieck–Riemann–Roch theorem from about 1956 is usually cited as the key moment for the introduction of this circle of ideas.