417 Sentences With "morphism" | Random Sentence Generator

In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a morphism of schemes.

Given a morphism of schemes f: X \to Y over a scheme S, the morphism X \to X \times_S Y induced by the identity 1_X : X \to X and f is called the graph morphism of f. The graph morphism of the identity is called the diagonal morphism.

This universal morphism consists of an object of and a morphism which contains projections.

In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism :\delta: X \to X \times_S X is a morphism determined by the universal property of the fiber product X \times_S X of p and p applied to the identity 1_X : X \to X and the identity 1_X. It is a special case of a graph morphism: given a morphism f: X \to Y over S, the graph morphism of it is X \to X \times_S Y induced by f and the identity 1_X. The diagonal embedding is the graph morphism of 1_X. By definition, X is a separated scheme over S (p: X \to S is a separated morphism) if the diagonal morphism is a closed immersion.

Infinite squarefree words can be generated by squarefree morphism. A morphism is called squarefree if the image of every squarefree word is squarefree. A morphism is called k–squarefree if the image of every squarefree word of length k is squarefree. Crochemore proves that a uniform morphism is squarefree if and only if it is 3-squarefree.

There is a more general notion of a smooth morphism of schemes, which is roughly a morphism with smooth fibers. In particular, a scheme X is smooth over a field k if and only if the morphism X → Spec k is smooth.

There are several different ways to define the Frobenius morphism for a scheme. The most fundamental is the absolute Frobenius morphism. However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme. There are several different ways of adapting the Frobenius morphism to the relative situation, each of which is useful in certain situations. Cartesian.

For any morphism of the form f: X \to F(A') in D, there exists a unique morphism h: A \to A' such that the following diagram commutes: The typical diagram of the definition of a universal morphism. We can dualize this categorical concept. A universal morphism from F to X is a unique pair (A, u: F(A) \to X) that satisfies the following universal property. For any morphism of the form f: F(A') \to X in D, there exists a unique morphism h: A' \to A such that the following diagram commutes: The most important arrow here is u: F(A) \to X which establishes the universal property.

Also, a morphism p: X \to S locally of finite presentation is an unramified morphism if and only if the diagonal embedding is an open immersion.

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).

Let and be partially ordered sets and let be an order-preserving map. The map is a bounded morphism (also known as p-morphism) if for each and , if , then there exists such that and . Theorem:Esakia (1974), Esakia (1985). The following conditions are equivalent: :(1) is a bounded morphism.

Suppose that there exists a closed immersion . If the morphism is an isomorphism, then p is a covering morphism for the cdh topology. The cd stands for completely decomposed (in the same sense it is used for the Nisnevich topology). An equivalent definition of a covering morphism is that it is a proper morphism p such that for any point x of the codomain, the fiber p−1(x) contains a point rational over the residue field of x.

For example, an absolute Frobenius morphism is a universal homeomorphism.

In algebraic geometry, a universal homeomorphism is a morphism of schemes f: X \to Y such that, for each morphism Y' \to Y, the base change X \times_Y Y' \to Y' is a homeomorphism of topological spaces. A morphism of schemes is a universal homeomorphism if and only if it is integral, radicial and surjective.EGA IV4, 18.12.11. In particular, a morphism of locally of finite type is a universal homeomorphism if and only if it is finite, radicial and surjective.

A morphism f:X\to S is called proper if # it is separated # of finite-type # universally closed The last condition means that given a morphism S' \to S the base change morphism S'\times_SX is a closed immersion. Most known examples of proper morphisms are in fact projective; but, examples of proper varieties which are not projective can be found using toric geometry.

Every morphism from a Noetherian scheme X \to S is quasi- compact.

Such a function is known as a morphism from ~A to ~B.

A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve X in the affine plane A2 defined by x2 = y3 is not normal, because there is a finite birational morphism A1 → X (namely, t maps to (t3, t2)) which is not an isomorphism. By contrast, the affine line A1 is normal: it cannot be simplified any further by finite birational morphisms.

Particular kinds of morphisms of groupoids are of interest. A morphism p: E \to B of groupoids is called a fibration if for each object x of E and each morphism b of B starting at p(x) there is a morphism e of E starting at x such that p(e)=b. A fibration is called a covering morphism or covering of groupoids if further such an e is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.

A morphism f: X → Y of schemes is called universally closed if for every scheme Z with a morphism Z → Y, the projection from the fiber product :X \times_Y Z \to Z is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ([EGA] II, 5.4.1 ). One also says that X is proper over Y. In particular, a variety X over a field k is said to be proper over k if the morphism X → Spec(k) is proper.

For the general definition, we start with a category C that has a terminal object, which we denote by 1. The object Ω of C is a subobject classifier for C if there exists a morphism :1 → Ω with the following property: :For each monomorphism j: U → X there is a unique morphism χ j: X → Ω such that the following commutative diagram center :is a pullback diagram—that is, U is the limit of the diagram: center The morphism χ j is then called the classifying morphism for the subobject represented by j.

Here kp denotes the image of k under the Frobenius morphism a→ap.

Some experts suspectpg 190 (see, for example, ) that triangulated categories are not really the "correct" concept. The essential reason is that the cone of a morphism is unique only up to a non-unique isomorphism. In particular, the cone of a morphism does not in general depend functorially on the morphism (note the non-uniqueness in axiom (TR 3), for example). This non- uniqueness is a potential source of errors.

150px In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f : X → Y and g : Y → X are morphisms whose composition f o g : Y → Y is the identity morphism on Y, then g is a section of f, and f is a retraction of g.Mac Lane (1978, p.19).

Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.

Given a category C and a morphism f\colon X\to Y in C , the image Section I.10 p.12 of f is a monomorphism m\colon I\to Y satisfying the following universal property: #There exists a morphism e\colon X\to I such that f = m\, e. #For any object I' with a morphism e'\colon X\to I' and a monomorphism m'\colon I'\to Y such that f = m'\, e', there exists a unique morphism v\colon I\to I' such that m = m'\, v. Remarks: # such a factorization does not necessarily exist.

Given a smooth scheme Y the projection morphism :Y\times X \to X is smooth.

Then a k-point of X means a k-point of Pn at which the given polynomials vanish. More generally, let X be a scheme over a field k. This means that a morphism of schemes f: X → Spec(k) is given. Then a k-point of X means a section of this morphism, that is, a morphism a: Spec(k) → X such that the composition fa is the identity on Spec(k).

Then a strongly representable morphism f: X \to Y of prestacks is said to have the property P if, for every morphism T \to Y, T a scheme viewed as a prestack, the induced projection X \times_Y T \to T has the property P.

Conversely, a monomorphism is called normal if it is the kernel of some morphism. A category is called normal if every monomorphism is normal. Abelian categories, in particular, are always normal. In this situation, the kernel of the cokernel of any morphism (which always exists in an abelian category) turns out to be the image of that morphism; in symbols: :im f = ker coker f (in an abelian category) When m is a monomorphism, it must be its own image; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know which morphism the monomorphism is a kernel of, to wit, its cokernel.

Hence, f is not quasi-compact. A morphism from a quasi-compact scheme to an affine scheme is quasi-compact. Let f: X \to Y be a quasi-compact morphism between schemes. Then f(X) is closed if and only if it is stable under specialization.

Some important properties P of morphisms of schemes are preserved under arbitrary base change. That is, if X → Y has property P and Z → Y is any morphism of schemes, then the base change X xY Z → Z has property P. For example, flat morphisms, smooth morphisms, proper morphisms, and many other classes of morphisms are preserved under arbitrary base change.. The word descent refers to the reverse question: if the pulled-back morphism X xY Z → Z has some property P, must the original morphism X → Y have property P? Clearly this is impossible in general: for example, Z might be the empty scheme, in which case the pulled- back morphism loses all information about the original morphism. But if the morphism Z → Y is flat and surjective (also called faithfully flat) and quasi- compact, then many properties do descend from Z to Y. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms.. These results form part of Grothendieck's theory of faithfully flat descent.

The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of 'complete', in the sense of 'no missing points', can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley.

This would involve the following: #Defining inverse object functions, checking that they are inverse, and checking that corresponding objects have the same underlying set. #Checking that a set function is "continuous" (i.e., a morphism) in the given category if and only if it is continuous (a morphism) in Top.

Fix a scheme S, called a base scheme. Then a morphism p: X \to S is called a scheme over S or an S-scheme; the idea of the terminology is that it is a scheme X together with a map to the base scheme S. For example, a vector bundle E → S over a scheme S is an S-scheme. An S-morphism from p:X →S to q:Y →S is a morphism ƒ:X →Y of schemes such that p = q ∘ ƒ. Given an S-scheme X \to S, viewing S as an S-scheme over itself via the identity map, an S-morphism S \to X is called a S-section or just a section.

In category theory, given a category C, an isomorphism is a morphism that has an inverse morphism , that is, and . For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

A morphism or homomorphism of posemigroups is a semigroup homomorphism that preserves the order (equivalently, that is monotonically increasing).

Given an excellent scheme X and a locally finite type morphism f:X'\to X, then X' is excellentpg 217.

A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.

A morphism between extensions of R by I, over say A, is an algebra homomorphism E → E that induces the identities on I and R. By the five lemma, such a morphism is necessarily an isomorphism, and so two extensions are equivalent if there is a morphism between them. Example: Let R be a commutative ring and M an R-module. Let E = R ⊕ M be the direct sum of abelian groups. Define the multiplication on E by :(a, x) \cdot (b, y) = (ab, ay + bx).

A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism. A category C is binormal if it's both normal and conormal. But note that some authors will use the word "normal" only to indicate that C is binormal.

A central problem in topology is determining when two spaces are the same i.e. homeomorphic or diffeomorphic. Constructing a morphism explicitly is almost always impractical. If we put further condition on one or both spaces (manifolds) we can exploit this additional structure in order to show that the desired morphism must exist.

In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points.

In homological algebra, Zeeman's comparison theorem, introduced by , gives conditions for a morphism of spectral sequences to be an isomorphism.

All species of Darwin's finches exhibit this morphism, which lasts for two months. No interpretation of this phenomenon is known.

In category theory, "map" is often used as a synonym for "morphism" or "arrow", and thus is more general than "function". For example, a morphism f:\, X \to Y in a concrete category (i.e. a morphism which can be viewed as functions) carries with it the information of its domain (the source X of the morphism) and its codomain (the target Y). In the widely used definition of a function f:X\to Y, f is a subset of X\times Y consisting of all the pairs (x,f(x)) for x\in X. In this sense, the function does not capture the information of which set Y is used as the codomain; only the range f(X) is determined by the function.

In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator). In a concrete category, an embedding is a morphism ƒ: A → B which is an injective function from the underlying set of A to the underlying set of B and is also an initial morphism in the following sense: If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒg: C → B, then g itself is a morphism. A factorization system for a category also gives rise to a notion of embedding.

It is the pair (A, u) which is essentially unique in this fashion. The object A itself is only unique up to isomorphism. Indeed, if (A, u) is a universal morphism and k: A \to A' is any isomorphism then the pair (A', u'), where u' = F(k) \circ u is also a universal morphism.

In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic. Put more precisely, we have a factorisation of f: A → B as :A → C → I → B, where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle (called the parallel of f) is an isomorphism. In a pre-abelian category, this is not necessarily true. The factorisation shown above does always exist, but the parallel might not be an isomorphism.

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.

A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. An isogeny is a finite-to-one morphism. When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case g = 1, the notion of abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for g > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus.

For example, the notion of restricting a morphism to a subset of its domain must be modified. For more, see subobject.

In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is injective (on geometric points). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem. Secondly, to other period mappings.

Typical examples of prime correspondences come from the graph \Gamma_f \subset X\times Y of a morphism of varieties f:X \to Y.

This also proves this morphism is not smooth from the equivalence between formally smooth morphisms locally of finite presentation and smooth morphisms.

A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism.

A morphism of chain complexes induces a morphism H_\bullet(F) of their homology groups, consisting of the homomorphisms H_n(F) : H_n(C) \to H_n(D) for all n. A morphism F is called a quasi-isomorphism if it induces an isomorphism on the nth homology for all n. Many constructions of chain complexes arising in algebra and geometry, including singular homology, have the following functoriality property: if two objects X and Y are connected by a map f, then the associated chain complexes are connected by a morphism F=C(f) : C_\bullet(X) \to C_\bullet(Y), and moreover, the composition g\circ f of maps f: X -> Y and g: Y -> Z induces the morphism C(g\circ f): C_\bullet(X) \to C_\bullet(Z) that coincides with the composition C(g) \circ C(f). It follows that the homology groups H_\bullet(C) are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.

An analytic space X which isn't reduced has a reduction Xred, a reduced analytic space with the same underlying topological space. There is a canonical morphism . Every morphism from X to a reduced analytic space factors through r. An analytic space is normal if every stalk of the structure sheaf is a normal ring (meaning an integrally closed integral domain).

In general, for an algebraic category C, an embedding between two C-algebraic structures X and Y is a C-morphism that is injective.

If the composition fg of two morphisms is an epimorphism, then f must be an epimorphism. As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context. If D is a subcategory of C, then every morphism in D that is an epimorphism when considered as a morphism in C is also an epimorphism in D. However the converse need not hold; the smaller category can (and often will) have more epimorphisms. As for most concepts in category theory, epimorphisms are preserved under equivalences of categories: given an equivalence F : C → D, a morphism f is an epimorphism in the category C if and only if F(f) is an epimorphism in D. A duality between two categories turns epimorphisms into monomorphisms, and vice versa.

The natural notion of morphism between Cauchy spaces is that of a Cauchy-continuous function, a concept that had earlier been studied for uniform spaces.

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 the mathematical field of category theory, specifically the theory of 2-categories, a lax natural transformation is a kind of morphism between 2-functors.

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties.

In mathematics, polyad is a concept of category theory introduced by Jean Bénabou in generalising monads. A polyad in a bicategory D is a bicategory morphism Φ from a locally punctual bicategory C to D, . (A bicategory C is called locally punctual if all hom-categories C(X,Y) consist of one object and one morphism only.) Monads are polyads where C has only one object.

Limits and colimits are defined as universal cones. That is, cones through which all other cones factor. A cone φ from L to F is a universal cone if for any other cone ψ from N to F there is a unique morphism from ψ to φ. center Equivalently, a universal cone to F is a universal morphism from Δ to F (thought of as an object in CJ), or a terminal object in (Δ ↓ F). Dually, a cone φ from F to L is a universal cone if for any other cone ψ from F to N there is a unique morphism from φ to ψ.

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 objects , , and of a category with finite products and coproducts, there is a canonical morphism , where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed): center The universal property of the product then guarantees a unique morphism induced by the dashed arrows in the above diagram. A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, one has the canonical isomorphism :X\times (Y + Z)\simeq (X\times Y)+ (X \times Z).

Since the structure sheaf of a scheme naturally carries only the discrete topology, the notion of formally étale for schemes is analogous to formally étale for the discrete topology for rings. That is, a morphism of schemes is formally étale if for every affine Y-scheme Z, every nilpotent sheaf of ideals J on Z with be the closed immersion determined by J, and every Y-morphism , there exists a unique Y-morphism such that .EGA IV4, Définition 17.1.1. It is equivalent to let Z be any Y-scheme and let J be a locally nilpotent sheaf of ideals on Z.EGA IV4, Remarques 17.1.

11th Int. Ornith. Cong., pp. 309–328. Basel. show various kinds of morphism. Males are dimorphic in song type: songs A and B are quite distinct.

In commutative algebra and algebraic geometry, a morphism is called formally étale if it has a lifting property that is analogous to being a local diffeomorphism.

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.

Let F : J → C be a diagram of shape J in a category C. A cone to F is an object N of C together with a family ψX : N → F(X) of morphisms indexed by the objects X of J, such that for every morphism f : X → Y in J, we have F(f) ∘ ψX = ψY. A limit of the diagram F : J → C is a cone (L, \varphi) to F such that for every other cone (N, ψ) to F there exists a unique morphism u : N → L such that \varphiX ∘ u = ψX for all X in J. A universal cone One says that the cone (N, ψ) factors through the cone (L, \varphi) with the unique factorization u. The morphism u is sometimes called the mediating morphism. Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information).

The normalization N(X) of an analytic space X comes with a canonical map . Every dominant morphism from a normal analytic space to X factors through ν.

The PTC gene, which accounts for 85% of the tasting variance, has now been analysed for sequence variation with results which suggest selection is maintaining the morphism.

Any reduced scheme X has a unique normalization: a normal scheme Y with an integral birational morphism Y → X. (For X a variety over a field, the morphism Y → X is finite, which is stronger than "integral".Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Corollary 13.13) The normalization of a scheme of dimension 1 is regular, and the normalization of a scheme of dimension 2 has only isolated singularities.

220px In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2.

Let M be a deterministic automaton with state space S and alphabet A. The words in the free monoid A∗ induce transformations of S giving rise to a monoid morphism from A∗ to the full transformation monoid TS. The image of this morphism is the transformation semigroup of M. For a regular language, the syntactic monoid is isomorphic to the transformation monoid of the minimal automaton of the language.

One advantage of defining varieties over arbitrary fields through the theory of schemes is that such definitions are intrinsic and free of embeddings into ambient affine n-space. A k-algebraic set is a separated and reduced scheme of finite type over Spec(k). A k-variety is an irreducible k-algebraic set. A k-morphism is a morphism between k-algebraic sets regarded as schemes over Spec(k).

That is, f is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of X to points in Y.

If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X : F(X) \to G(X) between objects of D . The morphism \eta_X is called the component of \eta at X . # Components must be such that for every morphism f :X \to Y in C we have: :::\eta_Y \circ F(f) = G(f) \circ \eta_X The last equation can conveniently be expressed by the commutative diagram This is the commutative diagram which is part of the definition of a natural transformation between two functors.

A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that: #E and M both contain all isomorphisms of C and are closed under composition. #Every morphism f of C can be factored as f=m\circ e for some morphisms e\in E and m\in M. #The factorization is functorial: if u and v are two morphisms such that vme=m'e'u for some morphisms e, e'\in E and m, m'\in M, then there exists a unique morphism w making the following diagram commute: center Remark: (u,v) is a morphism from me to m'e' in the arrow category.

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).

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.

More colloquially, this means that a morphism consists of a collection of maps F(i) \to G(j_i) for each i, where j_i is (depending on i) large enough.

A weak pullback of a cospan is a cone over the cospan that is only weakly universal, that is, the mediating morphism above is not required to be unique.

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.

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.

Every isomorphism is an epimorphism; indeed only a right-sided inverse is needed: if there exists a morphism j : Y → X such that fj = idY, then f: X → Y is easily seen to be an epimorphism. A map with such a right-sided inverse is called a split epi. In a topos, a map that is both a monic morphism and an epimorphism is an isomorphism. The composition of two epimorphisms is again an epimorphism.

Accordingly, true refers to the element 1, which is selected by the arrow: true: {0} → {0, 1} that maps 0 to 1. The subset A of S can now be defined as the pullback of true along the characteristic function χA, shown on the following diagram: center Defined that way, χ is a morphism SubC(S) → HomC(S, Ω). By definition, Ω is a subobject classifier if this morphism is an isomorphism.

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 .

In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational. Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected.

Dually is the notion of cofibrant object, defined to be an object c such that the unique morphism \varnothing\to c from the initial object to c is a cofibration.

If X and Y are topological spaces and u is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi.

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).

While these matrices are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object. Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objects and in an additive category, there is exactly one morphism from to 0 (just as there is exactly one 0-by-1 matrix with entries in ) and exactly one morphism from 0 to (just as there is exactly one 1-by-0 matrix with entries in ) – this is just what it means to say that 0 is a zero object. Furthermore, the zero morphism from to is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.

Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition: :The object B is a biproduct of the objects A1, ..., An if and only if there are projection morphisms pj: B → Aj and injection morphisms ij: Aj → B, such that (i1∘p1) \+ ··· \+ (in∘pn) is the identity morphism of B, pj∘ij is the identity morphism of Aj, and pj∘ik is the zero morphism from Ak to Aj whenever j and k are distinct. This biproduct is often written A1 ⊕ ··· ⊕ An, borrowing the notation for the direct sum. This is because the biproduct in well known preadditive categories like Ab is the direct sum.

A representation of a quiver Q is an association of an R-module to each vertex of Q, and a morphism between each module for each arrow. A representation V of a quiver Q is said to be trivial if V(x) = 0 for all vertices x in Q. A morphism, f: V → V′, between representations of the quiver Q, is a collection of linear maps f(x):V(x)\rightarrow V'(x) such that for every arrow a in Q from x to y V'(a) f(x) = f(y) V(a) , i.e. the squares that f forms with the arrows of V and V′ all commute. A morphism, f, is an isomorphism, if f(x) is invertible for all vertices x in the quiver.

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.

In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces.

Not all mathematical structures are F-algebras. For example, a poset P may be defined in categorical terms with a morphism s:P × P -> Ω, on a subobject classifier (Ω = {0,1} in the category of sets and s(x,y)=1 precisely when x≤y). The axioms restricting the morphism s to define a poset can be rewritten in terms of morphisms. However, as the codomain of s is Ω and not P, it is not an F-algebra.

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.

In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); # C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; # given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f). Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.

A morphism in an allegory is called a map if it is entire (1\subseteq R^\circ R) and deterministic (RR^\circ \subseteq 1). Another way of saying this is that a map is a morphism that has a right adjoint in when is considered, using the local order structure, as a 2-category. Maps in an allegory are closed under identity and composition. Thus, there is a subcategory of with the same objects but only the maps as morphisms.

These branes can be viewed as objects with a morphism between them. In this case, the morphism will be the state of a string that stretches between brane A and brane B. Singularities are avoided because the observed consequences of "Big Crunches" never reach zero size. In fact, should the universe begin a "big crunch" sort of process, string theory dictates that the universe could never be smaller than the size of one string, at which point it would actually begin expanding.

In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. In general, it is the map which preserves the widest amount of structure, and it tends to be unique. In the rare cases where latitude in choices remains, the map is either conventionally agreed upon to be the most useful for further analysis, or sometimes the most elegant map known up to date. A standard form of canonical map involves some function mapping a set X to the set X/R (X modulo R), where R is an equivalence relation on X. A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object.

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.

In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper.

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.

In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism. In preadditive categories it makes sense to add and subtract morphisms (the hom- sets actually form abelian groups). In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference: :coeq(f, g) = coker(g – f). A stronger notion is that of an absolute coequalizer, this is a coequalizer that is preserved under all functors.

In category theory, n-ary functions generalise to n-ary morphisms in a multicategory. The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category. The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.

In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (a.k.a group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism f : A → B of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.

In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object.

The typical diagram of the definition of a universal morphism. In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism (see Formal Definition). Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category (see Connection with Comma Categories). Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

Direct systems depend on the notion of filtered categories. For example, the category N, whose objects are natural numbers, and with exactly one morphism from n to m whenever n \le m, is a filtered category.

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.

Suppose that is a finitely presented morphism of affine schemes, s is a point of S, and M is a finite type OX-module. If n is a natural number, then Gruson and Raynaud define an S-dévissage in dimension n to consist of: # A closed finitely presented subscheme X′ of X containing the closed subscheme defined by the annihilator of M and such that the dimension of is less than or equal to n. # A scheme T and a factorization of the restriction of f to X′ such that is a finite morphism and is a smooth affine morphism with geometrically integral fibers of dimension n. Denote the generic point of by τ and the pushforward of M to T by N. # A free finite type OT-module L and a homomorphism such that is bijective.

Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category. When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A. However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.

For categories arising from partially ordered sets (P, \le) (with a single morphism from x to y iff x \le y), then the formalism becomes much simpler: adjoint pairs are Galois connections and monads are closure operators.

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 notion of group action can be put in a broader context by using the action groupoid G'=G \ltimes X associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. For more details, see the book Topology and groupoids referenced below. This action groupoid comes with a morphism p: G′ → G which is a covering morphism of groupoids.

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism f : X → Y is the "most general" morphism k : K → X that yields zero when composed with (followed by) f. Note that kernel pairs and difference kernels (also known as binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article.

In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism \pi: X \to Y that :(i) is invariant; i.e., \pi \circ \sigma = \pi \circ p_2 where \sigma: G \times X \to X is the given group action and p2 is the projection. :(ii) satisfies the universal property: any morphism X \to Z satisfying (i) uniquely factors through \pi. One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes.

The set of events can then be structured in the same way as invariance of causal structure, or local-to- global causal connections or even formal properties of global causal connections. The morphisms between (non-trivial) objects could be viewed as representing causal connections leading from one event to another one. For example, the morphism f above leads form event s1 to event s2. The sequences or "paths" of morphisms for which there is no inverse morphism, could then be interpreted as defining horismotic or chronological precedence relations.

In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.

The action of A on R is given by: :c \cdot \sum a_\alpha X^\alpha = \sum c a_\alpha X^\alpha, where α is a multi-index. Let . Then is the affine scheme , but its structure morphism , and hence the action of A on R, is different: :c \cdot \sum a_\alpha X^\alpha = \sum F(c) a_\alpha X^\alpha = \sum c^p a_\alpha X^\alpha. Because restriction of scalars by Frobenius is simply composition, many properties of are inherited by XF under appropriate hypotheses on the Frobenius morphism.

Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative). In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if f : X → Y is a split epimorphism with split monomorphism g : Y → X, then X is isomorphic to the direct sum of Y and the kernel of f. The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work.

The n-th term of the period-doubling sequence d(n) () is determined by the parity of the exponent of the highest power of 2 dividing n. It is also the fixed point of the morphism 0 → 01, 1 → 00.Allouche & Shallit (2003) p. 176 Starting with the initial term w = 0 and iterating the 2-uniform morphism φ on w where φ(0) = 01 and φ(1) = 00, it is evident that the period-doubling sequence is the fixed-point of φ(w) and thus it is 2-automatic.

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.

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 .

This definition is closely related to the notion of a formally smooth morphism of schemes. If in addition the map between the tangent spaces of F and G is an isomorphism, then F is called a hull of G.

London: Chapman & Hall and for classical genetics by John Maynard Smith (1998).Smith, John Maynard. 1998. Evolutionary Genetics (2nd ed.). Oxford: Oxford U. Pr. The shorter term morphism may be more accurate than polymorphism, but is not often used.

A subgroup H of a group G determines a covering morphism of groupoids p: K \rightarrow G and if X is a generating set for G then its inverse image under p is the Schreier graph of (G,X).

In algebraic geometry, the dimension of a scheme is a generalization of a dimension of an algebraic variety. Scheme theory emphasizes the relative point of view and, accordingly, the relative dimension of a morphism of schemes is also important.

In mathematics, the Prym variety construction (named for Friedrich Prym) is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by F. Schottky and H. W. E. Jung in relation with the Schottky problem, as it is now called, of characterising Jacobian varieties among abelian varieties. It is said to have appeared first in the late work of Riemann, and was extensively studied by Wirtinger in 1895, including degenerate cases. Given a non-constant morphism :φ: C1 -> C2 of algebraic curves, write Ji for the Jacobian variety of Ci. Then from φ construct the corresponding morphism :ψ: J1 -> J2, which can be defined on a divisor class D of degree zero by applying φ to each point of the divisor.

The map L → U(L) of K-modules canonically extends to a map T(L) → U(L) of algebras, where T(L) is the tensor algebra on L (for example, by the universal property of tensor algebras), and this is a filtered map equipping T(L) with the filtration putting L in degree one (actually, T(L) is graded). Then, passing to the associated graded, one gets a canonical morphism T(L) → grU(L), which kills the elements vw - wv for v, w ∈ L, and hence descends to a canonical morphism S(L) → grU(L). Then, the (graded) PBW theorem can be reformulated as the statement that, under certain hypotheses, this final morphism is an isomorphism of commutative algebras. This is not true for all K and L (see, for example, the last section of Cohn's 1961 paper), but is true in many cases.

A morphism of magmas is a function, , mapping magma M to magma N, that preserves the binary operation: :f (x •M y) = f(x) •N f(y) where •M and •N denote the binary operation on M and N respectively.

Since groups are categories, one can also consider representation of other categories. The simplest generalization is to monoids, which are categories with one object. Groups are monoids for which every morphism is invertible. General monoids have representations in any category.

In algebraic geometry, the syntomic topology is a Grothendieck topology introduced by . Mazur defined a morphism to be syntomic if it is flat and locally a complete intersection. The syntomic topology is generated by surjective syntomic morphisms of affine schemes.

Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not separated. The condition that a morphism is quasi-separated often occurs together with the condition that it is quasi- compact.

Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, while the monomorphism is called the image of f.

A birational map from X to Y is a rational map f: X ⇢ Y such that there is a rational map Y ⇢ X inverse to f. A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y. In this case, X and Y are said to be birational, or birationally equivalent. In algebraic terms, two varieties over a field k are birational if and only if their function fields are isomorphic as extension fields of k. A special case is a birational morphism f: X → Y, meaning a morphism which is birational.

There is a way to associate a stack to a given prestack. It is similar to the sheafification of a presheaf and is called stackification. The idea of the construction is quite simple: given a prestack p: F \to C, we let HF be the category where an object is a descent datum and a morphism is that of descent data. (The details are omitted for now) As it turns out, it is a stack and comes with a natural morphism \theta: F \to HF such that F is a stack if and only if θ is an isomorphism.

Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In particular, it is true in the categories of all groups, of all rings, and in any abelian category.

In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.John C. Baez and Mike Stay, "Physics, Topology, Logic and Computation: A Rosetta Stone", (2009) ArXiv 0903.0340 in New Structures for Physics, ed.

In Cartesian closed categories, a "function of two variables" (a morphism f : X×Y → Z) can always be represented as a "function of one variable" (the morphism λf : X → ZY). In computer science applications, this is known as currying; it has led to the realization that simply-typed lambda calculus can be interpreted in any Cartesian closed category. The Curry–Howard–Lambek correspondence provides a deep isomorphism between intuitionistic logic, simply-typed lambda calculus and Cartesian closed categories. Certain Cartesian closed categories, the topoi, have been proposed as a general setting for mathematics, instead of traditional set theory.

Explicitly, a morphism from a variety X to projective space Pn over a field k determines a line bundle L on X, the pullback of the standard line bundle O(1) on Pn. Moreover, L comes with n+1 sections whose base locus (the intersection of their zero sets) is empty. Conversely, any line bundle L with n+1 global sections whose common base locus is empty determines a morphism X → Pn.Hartshorne (1977), Theorem II.7.1. These observations lead to several notions of positivity for Cartier divisors (or line bundles), such as ample divisors and nef divisors.Lazarsfeld (2004), Chapter 1.

A curve , over is called a modular curve if for some there exists a surjective morphism , given by a rational map with integer coefficients. The famous modularity theorem tells us that all elliptic curves over are modular. Mappings also arise in connection with since points on it correspond to some -isogenous pairs of elliptic curves. An isogeny between two elliptic curves is a non-trivial morphism of varieties (defined by a rational map) between the curves which also respects the group laws, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity.

Over the complex numbers, a polarised abelian variety can also be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H1 and H2 are called equivalent if there are positive integers n and m such that nH1=mH2. A choice of an equivalence class of Riemann forms on A is called a polarisation of A. A morphism of polarised abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A.

Suppose that R is a Dedekind domain with field of fractions K, and suppose that AK is a smooth separated scheme over K (such as an abelian variety). Then a Néron model of AK is defined to be a smooth separated scheme AR over R with fiber AK that is universal in the following sense. :If X is a smooth separated scheme over R then any K-morphism from XK to AK can be extended to a unique R-morphism from X to AR (Néron mapping property). In particular, the canonical map A_R(R)\to A_K(K) is an isomorphism.

A pomonoid can be considered as a monoidal category that is both skeletal and thin, with an object of for each element of , a unique morphism from to if and only if , the tensor product being given by , and the unit by .

This equivalence relation is known as the kernel of . More generally, a function may map equivalent arguments (under an equivalence relation on ) to equivalent values (under an equivalence relation on ). Such a function is a morphism of sets equipped with an equivalence relation.

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).

It therefore has a foundational role in syntax. A free magma has the universal property such that, if is a function from X to any magma, N, then there is a unique extension of f to a morphism of magmas, f ′ : f ′ : MX → N.

In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism. A normal category is a category in which every monomorphism is normal. A conormal category is one in which every epimorphism is conormal.

A morphism of spectral sequences E → E' is by definition a collection of maps fr : Er → E'r which are compatible with the differentials and with the given isomorphisms between cohomology of the rth step and the (r+1)th sheets of E and E' , respectively.

It follows that Field is not a reflective subcategory of CRing. The category of fields is neither finitely complete nor finitely cocomplete. In particular, Field has neither products nor coproducts. Another curious aspect of the category of fields is that every morphism is a monomorphism.

This is one instance of the geometric unsuitability of the Zariski topology. Grothendieck solved this problem by defining the notion of properness of a scheme (actually, of a morphism of schemes), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.

3 of EGA III quoted above is that if f:X→Y is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over Y. Then structure theorem for quasi-finite morphisms applies and yields the desired result.

However, although infinite direct sums make sense in some categories, like Ab, infinite biproducts do not make sense. The biproduct condition in the case n = 0 simplifies drastically; B is a nullary biproduct if and only if the identity morphism of B is the zero morphism from B to itself, or equivalently if the hom-set Hom(B,B) is the trivial ring. Note that because a nullary biproduct will be both terminal (a nullary product) and initial (a nullary coproduct), it will in fact be a zero object. Indeed, the term "zero object" originated in the study of preadditive categories like Ab, where the zero object is the zero group.

In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, given a group homomorphism f : G → H, we can define the group K = im(f) and then write f as the composition of the surjective homomorphism G → K that is defined like f, followed by the injective homomorphism K → H that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in (though not in all concrete categories).

Given a locally ringed space (X, OX), certain sheaves of modules on X occur in the applications, the OX-modules. To define them, consider a sheaf F of abelian groups on X. If F(U) is a module over the ring OX(U) for every open set U in X, and the restriction maps are compatible with the module structure, then we call F an OX-module. In this case, the stalk of F at x will be a module over the local ring (stalk) Rx, for every x∈X. A morphism between two such OX-modules is a morphism of sheaves which is compatible with the given module structures.

Category theory deals with morphisms instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned—or at least formulated more abstractly.

The most general comma category construction involves two functors with the same codomain. Often one of these will have domain 1 (the one-object one-morphism category). Some accounts of category theory consider only these special cases, but the term comma category is actually much more general.

Orthogonal projection onto a line, , is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism.

The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism f : G \rightarrow H. The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into \iota \circ \pi, where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object \ker f and a monomorphism \kappa: \ker f \rightarrow G (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from \ker f to H and G / \ker f.

Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, every morphism has a kernel. The category of associative rngs (not rings!) together with multiplicative morphisms is pseudo-abelian. A more complicated example is the category of Chow motives.

In mathematics the cotangent complex is roughly a universal linearization of a morphism of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M. André independently came up with a definition that works in all cases.

Let be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms and . By composing these with f, we construct two morphisms: :, and :. Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement: :.

A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N(x)β/2, and is called mixed of weight ≤β if it can be written as repeated extensions by pure sheaves with weights ≤β. Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then Rif! takes mixed sheaves of weight ≤β to mixed sheaves of weight ≤β+i. The original Weil conjectures follow by taking f to be a morphism from a smooth projective variety to a point and considering the constant sheaf Ql on the variety.

The pair (E,M) of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions: #Every morphism f of C can be factored as f=m\circ e with e\in E and m\in M. #E=M^\uparrow and M=E^\downarrow.

In mathematics, the permutation category is a category where # an object is a natural number, # a morphism n \to m is an element of the symmetric group \Sigma_n when n = m and is none otherwise. It is equivalent as an category to the category of finite sets and bijections between them.

This notion of inverse also readily generalizes to categories. An inverse category is simply a category in which every morphism f:X→Y has a generalized inverse g:Y→X such that fgf = f and gfg = g. An inverse category is selfdual. The category of sets and partial bijections is the prime example.

Another example of a pullback comes from the theory of fiber bundles: given a bundle map and a continuous map , the pullback (formed in the category of topological spaces with continuous maps) is a fiber bundle over called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.

The morphism part is given by mapping a group homomorphism to the function , which goes from to . ;Remarks: #Properties (Dl) and (Dr) express biadditivity of φ, which may be regarded as distributivity of φ over addition. #Property (A) resembles some associative property of φ. #Every ring R is an R-bimodule.

The structures given by composition are axiomatized and generalized in category theory with the concept of morphism as the category-theoretical replacement of functions. The reversed order of composition in the formula applies for composition of relations using converse relations, and thus in group theory. These structures form dagger categories.

If C and D are categories, one can form the product category C × D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairs can be composed componentwise.

Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories. Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian.

Another theorem of his concerns the constructible sets in algebraic geometry, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states that the image of such a set by a morphism of algebraic varieties is of the same type. Logicians call this an elimination of quantifiers.

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y. More explicitly, a coequalizer can be defined as an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g. Moreover, the pair (Q, q) must be universal in the sense that given any other such pair (Q′, q′) there exists a unique morphism u : Q → Q′ such that u ∘ q = q′. This information can be captured by the following commutative diagram: Image:Coequalizer-01.png As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).

For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds, reflecting as for Riemann surfaces that ramify at single points that it happens in real codimension two). A classical result, Zariski-Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular algebraic variety a branch locus, namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor). There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possible "open subsets of failure" to be an étale morphism.

Suppose that A is a complete discrete valuation ring of characteristic 0 with quotient field k of characteristic p>0 and perfect. An affine enlargement of a scheme X0 over k consists of a torsion-free A-algebra B and an ideal I of B such that B is complete in the I topology and the image of I is nilpotent in B/pB, together with a morphism from Spec(B/I) to X0. A convergent isocrystal over a k-scheme X0 consists of a module over B⊗Q for every affine enlargement B that is compatible with maps between affine enlargements . An F-isocrystal (short for Frobenius isocrystal) is an isocrystal together with an isomorphism to its pullback under a Frobenius morphism.

In algebraic geometry, the term branched covering is used to describe morphisms f from an algebraic variety V to another one W, the two dimensions being the same, and the typical fibre of f being of dimension 0. In that case, there will be an open set W' of W (for the Zariski topology) that is dense in W, such that the restriction of f to W' (from V' = f^{-1}(W') to W', that is) is unramified. Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness and separability). Generically, then, such a morphism resembles a covering space in the topological sense.

In algebraic geometry, a morphism of schemes f from X to Y is called quasi- separated if the diagonal map from X to X×YX is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi compact). A scheme X is called quasi-separated if the morphism to Spec Z is quasi-separated. Quasi- separated algebraic spaces and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that X is quasi-separated as part of the definition of an algebraic space or algebraic stack X. Quasi-separated morphisms were introduced by as a generalization of separated morphisms. All separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated.

In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice. In the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis. In the category of abelian groups, the epimorphism Z → Z/2Z which sends every integer to its remainder modulo 2 does not split; in fact the only morphism Z/2Z → Z is the zero map. Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z.

DGD, Dworkin's Game Driver (at one time called Dworkin's Generic Driver), is an LPMud server written by Felix A. "Dworkin" Croes. DGD pioneered important technical innovations in MUDs, particularly disk-based object storage, full world persistence, separation of concerns between driver and mudlib, runtime morphism, automatic garbage collection, lightweight objects and LPC-to-C compilation.

The above layman's description can be stated more formally in category theory: the anamorphism of a coinductive type denotes the assignment of a coalgebra to its unique morphism to the final coalgebra of an endofunctor. These objects are used in functional programming as unfolds. The categorical dual (aka opposite) of the anamorphism is the catamorphism.

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).

In a different direction, it is finer than the qfh topology, so h locally, algebraic correspondences are finite sums of morphisms.Suslin, Voevodsky, Singular homology of abstract algebraic varieties Finally, every proper surjective morphism is an h covering, so in any situation where de Jong's theorem on alterations is valid, h locally all schemes are regular.

The terms "abuse of language" and "abuse of notation" depend on context. Writing "f: A → B" for a partial function from A to B is almost always an abuse of notation, but not in a category theoretic context, where f can be seen as a morphism in the category of sets and partial functions.

Example: for any field extension k ⊂ E, the morphism Spec(E) → Spec(k) is faithfully flat and quasi-compact. So the descent results mentioned imply that a scheme X over k is smooth over k if and only if the base change XE is smooth over E. The same goes for properness and many other properties.

In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.

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, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel .Artin, 1972, p. 413. Recall that an idempotent morphism p is an endomorphism of an object with the property that p\circ p = p. Elementary considerations show that every idempotent then has a cokernel.

The dual concept to that of kernel is that of cokernel. That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. As mentioned above, a kernel is a type of binary equaliser, or difference kernel. Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel.

A morphism of schemes f:X \to Y is called étale if it is flat and unramfied. These are the algebro-geometric analogue of covering spaces. The two main examples to think of are covering spaces and finite separable field extensions. Examples in the first case can be constructed by looking at branched coverings and restricting to the unramified locus.

While defined on all schemes, the h and qfh topology are only ever used on Noetherian schemes. The h topology has various non-equivalent extensions to non-Noetherian schemes including the ph topologyA cohomological bound for the h-topology and the v topology. The proper cdh topology is defined as follows. Let be a proper morphism.

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.

As noted several times before, pt and Ω usually are not inverses. In general neither is X homeomorphic to pt(Ω(X)) nor is L order-isomorphic to Ω(pt(L)). However, when introducing the topology of pt(L) above, a mapping φ from L to Ω(pt(L)) was applied. This mapping is indeed a frame morphism.

The Auslander-Reiten quiver of an Artin algebra has a vertex for each indecomposable module and an arrow between vertices if there is an irreducible morphism between the corresponding modules. It has a map τ = D Tr called the translation from the non-projective vertices to the non-injective vertices, where D is the dual and Tr the transpose.

Hilbert's Nullstellensatz of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of the Nullstellensatz states that if R is a Jacobson ring, then so is any finitely generated R-algebra S. Moreover, the pullback of any maximal ideal J of S is a maximal ideal I of R, and S/J is a finite extension of the field R/I. In particular a morphism of finite type of Jacobson rings induces a morphism of the maximal spectrums of the rings. This explains why for algebraic varieties over fields it is often sufficient to work with the maximal ideals rather than with all prime ideals, as was done before the introduction of schemes.

A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors. If F and G are (covariant) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative: Commutative diagram defining natural transformations The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that ηX is an isomorphism for every object X in C.

The graph K is called invariant or sometimes the gluing graph. A rewriting step or application of a rule r to a host graph G is defined by two pushout diagrams both originating in the same morphism k\colon K\rightarrow D, where D is a context graph (this is where the name double-pushout comes from). Another graph morphism m\colon L\rightarrow G models an occurrence of L in G and is called a match. Practical understanding of this is that L is a subgraph that is matched from G (see subgraph isomorphism problem), and after a match is found, L is replaced with R in host graph G where K serves as an interface, containing the nodes and edges which are preserved when applying the rule.

In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point.

A differential graded algebra (or simply DG-algebra) A is a graded algebra equipped with a map d\colon A \to A which has either degree 1 (cochain complex convention) or degree -1 (chain complex convention) that satisfies two conditions: A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the monoidal category of chain complexes. A DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential d. A differential graded augmented algebra (also called a DGA- algebra, an augmented DG-algebra or simply a DGA) is a DG-algebra equipped with a DG morphism to the ground ring (the terminology is due to Henri Cartan). Warning: some sources use the term DGA for a DG-algebra.

In computer science, more precisely in automata theory, a recognizable set of a monoid is a subset that can be distinguished by some morphism to a finite monoid. Recognizable sets are useful in automata theory, formal languages and algebra. This notion is different from the notion of recognizable language. Indeed, the term "recognizable" has a different meaning in computability theory.

His literary fictions are often dense with allusions. Labyrinthine annotations were added to "Uyūshitan" when it was published in book form in 2009, where there were none when published initially in literary magazine. Often, his science fiction works take motif from mathematics. The narrator of "Boy's Surface" (2007) is a morphism, and the title is a reference to a geometrical notion.

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.

In this interpretation the cut rule of the sequent calculus corresponds to composition in the category. Boolean and Heyting algebras enter this picture as special categories having at most one morphism per homset, i.e., one proof per entailment, corresponding to the idea that existence of proofs is all that matters: any proof will do and there is no point in distinguishing them.

This is a well-defined morphism, often called the norm homomorphism. Then the Prym variety of φ is the kernel of ψ. To qualify that somewhat, to get an abelian variety, the connected component of the identity of the reduced scheme underlying the kernel may be intended. Or in other words take the largest abelian subvariety of J1 on which ψ is trivial.

In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model structure). It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism. The homotopy hypothesis states that ∞-groupoids are spaces.

Given a developing map \varphi, the monodromy or holonomy of a (G,X)-structure is the unique morphism h : \pi_1(M) \to G which satisfies : \forall \gamma \in \pi_1(M), p\in \tilde M : \varphi(\gamma\cdot p) = h(\gamma)\cdot \varphi(p). It depends on the choice of a developing map but only up to an inner automorphism of G.

In category theory, a weak inverse of an object A in a monoidal category C with monoidal product ⊗ and unit object I is an object B such that both and are isomorphic to the unit object I of C. A monoidal category in which every morphism is invertible and every object has a weak inverse is called a 2-group.

Here the objects correspond to the elements of P, and there is one morphism for objects which are related, zero otherwise. Alternately, a preordered set can be understood as an enriched category, enriched over the category . A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preordered class.

On the other hand, if it is symmetric, that is, if implies , then it is an equivalence relation. A preorder is total if or for all a, b. Equivalently, the notion of a preordered set P can be formulated in a categorical framework as a thin category; i.e., as a category with at most one morphism from an object to another.

A unit in an allegory is an object for which the identity is the largest morphism U\to U, and such that from every other object, there is an entire relation to . An allegory with a unit is called unital. Given a tabular allegory , the category is a regular category (it has a terminal object) if and only if is unital.

An R-algebra A is finite if it is finitely generated as an R-module. An R-algebra can be thought as a homomorphism of rings f\colon R\to A, in this case f is called a finite morphism if A is a finite R-algebra. The definition of finite algebra is related to that of algebras of finite type.

The same is true for terminal objects. For complete categories there is an existence theorem for initial objects. Specifically, a (locally small) complete category has an initial object if and only if there exist a set ( a proper class) and an -indexed family of objects of such that for any object of , there is at least one morphism for some .

The predators are more quickly to adapt and decrease apostatic selection when a drastic and abrupt change to the prey frequencies occur. This does not change the flexibility of the predators, but elicits a very high speed in the change of the search image. Apostatic selection is most strong in environments in which the prey with the rare morphism match the background.

The composition of quasi- compact morphisms is quasi-compact. The base change of a quasi-compact morphism is quasi-compact. An affine scheme is quasi-compact. In fact, a scheme is quasi-compact if and only if it is a finite union of open affine subschemes. Serre’s criterion gives a necessary and sufficient condition for a quasi-compact scheme to be affine.

Recall that a valuation ring A is a domain, so if K is the field of fractions of A, then Spec K is the generic point of Spec A. Let X and Y be schemes, and let f : X -> Y be a morphism of schemes. Then the following are equivalent:EGA II, proposition 7.2.3 and théorème 7.3.8.Stacks Project, tags 01KA, 01KY, and 0BX4.

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.

These concepts generalize respectively those of preordered set, partially ordered set and totally ordered set. However, it is difficult to work with them as in the small case because many constructions common in a set theory are no longer possible in this framework. Equivalently, a preordered class is a thin category, that is, a category with at most one morphism from an object to another.

The bracket is not skew- symmetric as one can see from the third axiom. Instead it fulfills a certain Jacobi-identity (first axiom) and a Leibniz rule (second axiom). From these two axioms one can derive that the anchor map ρ is a morphism of brackets: :: \rho[\phi,\psi] = [\rho(\phi),\rho(\psi)] . The fourth rule is an invariance of the inner product under the bracket.

EGA IV2, Corollaire 6.9.3 Suppose that S is a noetherian scheme, is a finite type morphism, and F is a coherent OX module. Then there exists a partition of S into locally closed subsets S1, ..., Sn with the following property: Give each Si its reduced scheme structure, denote by Xi the fiber product , and denote by Fi the restriction ; then each Fi is flat.

This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece. There is an equivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras.

Alan Artner compared the technique to Bacon's appropriation of Monet-like painterliness in his disquieting works; Buzz Spector felt this joy in painting offset a potentially moralistic tone in the work.Spector, Buzz. "Paul Lamantia," Artforum, January 1987. Lamantia's figuration also evolved towards an H.P. Lovecraftian, sci-fi- like "mechano-morphism," with "metallic musculature suggesting malevolent sexual machinery" merging with his existing re-grafted mutations.

In mathematics, a 2-valued morphism. is a homomorphism that sends a Boolean algebra B onto the two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on B, and, in a different way, also the same things as a maximal ideal of B. 2-valued morphisms have also been proposed as a tool for unifying the language of physics.

In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre also a projective line. Then the theorem states that S is rational.

In a categorical definition,Burgin (2011) , p. 57–69 named sets are built inside a chosen (mathematical) category similar to the construction of set theory in a topos. Namely, given a category K, a named set in K is a triad X = (X, f, I), in which X and I are two objects from K and f is a morphism between X and I.

Let A be an object in the concrete category C. Then A has the fixed-point property if every morphism (i.e., every function) f: A \to A has a fixed point. The most common usage is when C = Top is the category of topological spaces. Then a topological space X has the fixed-point property if every continuous map f: X \to X has a fixed point.

Therefore, for any c in C, exists k in ker t such that c = r(k), and r(ker t) = C. If , then is in ; since the intersection of and , then . Therefore, the restriction of the morphism is an isomorphism; and is isomorphic to . Finally, is isomorphic to due to the exactness of ; so B is isomorphic to the direct sum of and , which proves (3).

The alternative diagram representation of the second definition of a universal morphism. Of course, the diagrams are the same; choosing which way to write it is a matter of taste. They simply differ by a counterclockwise rotation of 180 degrees. However, the original diagram is preferable, because it illustrates the duality between the two definitions, as it is clear the arrows are being reversed in each case.

In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by . The word "Frobenioid" is a portmanteau of Frobenius and monoid, as certain Frobenius morphisms between Frobenioids are analogues of the usual Frobenius morphism, and some of the simplest examples of Frobenioids are essentially monoids.

10Honkala (2010) p.505 In general, a morphic word is the image of a pure morphic word under a coding. If a morphic word is constructed as the fixed point of a prolongable k-uniform morphism on A∗ then the word is k-automatic. The n-th term in such a sequence can be produced by a finite state automaton reading the digits of n in base k.

Eisenbud (1995), Corollary 18.17. A geometric reformulation is as follows. Let X be a connected affine scheme of finite type over a field K (for example, an affine variety). Let n be the dimension of X. By Noether normalization, there is a finite morphism f from X to affine space An over K. Then X is Cohen–Macaulay if and only all fibers of f have the same degree.

Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category J). Dually, a category is finitely cocomplete if all finite colimits exist.

For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning as well as another one. In category theory, a map may refer to a morphism, which is a generalization of the idea of a function. In some occasions, the term transformation can also be used interchangeably. There are also a few less common uses in logic and graph theory.

Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The weak factorization theorem says that every birational map can be factored as a composition of particularly simple blowups. The Cremona group, the group of birational automorphisms of the plane, is generated by blowups. Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces.

In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator). The identity morphism (identity mapping) is called the trivial automorphism in some contexts.

In category theory, an automorphism is an endomorphism (i.e., a morphism from an object to itself) which is also an isomorphism (in the categorical sense of the word). This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.

Preimages of sets under functions can be described as pullbacks as follows: Suppose , . Let be the inclusion map . Then a pullback of and (in ) is given by the preimage together with the inclusion of the preimage in : and the restriction of to :. Because of this example, in a general category the pullback of a morphism and a monomorphism can be thought of as the "preimage" under of the subobject specified by .

Every poset (and every preordered set) may be considered as a category where, for objects x and y, there is at most one morphism from x to y. More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Such categories are sometimes called posetal. Posets are equivalent to one another if and only if they are isomorphic.

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.

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

In algebraic geometry, a Fano fibration or Fano fiber space, named after Gino Fano, is a morphism of varieties whose general fiber is a Fano variety (in other words has ample anticanonical bundle) of positive dimension. The ones arising from extremal contractions in the minimal model program are called Mori fibrations or Mori fiber spaces (for Shigefumi Mori). They appear as standard forms for varieties without a minimal model.

The significance of Grothendieck's approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas Grothendieck saw it as a theorem about a morphism between varieties. By finding the right generalization, the proof became simpler while the conclusion became more general. In short, Grothendieck applied a strong categorical approach to a hard piece of analysis.

The selection maintaining the polymorphism maximises the species' niche by expanding its feeding opportunity. The genetics of this situation cannot be clarified in the absence of a detailed breeding program, but two loci with linkage disequilibrium is a possibility. Another interesting dimorphism is for the bills of young finches, which are either 'pink' or 'yellow'. All species of Darwin's finches exhibit this morphism, which lasts for two months.

In the mathematical discipline of category theory, a strict initial object is an initial object 0 of a category C with the property that every morphism in C with codomain 0 is an isomorphism. If C is a Cartesian closed category, then any initial object 0 of C is strict. Also, if C is a distributive or extensive category, then the initial object 0 of C is strict.

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 mathematics, specifically category theory, an essential monomorphism is a monomorphism f in a category C such that for a morphism g in C, g \circ f is a monomorphism only when g is a monomorphism. Essential monomorphisms in a category of modules are those whose image is an essential submodule of the codomain. An injective hull of an object X is an essential monomorphism from X to an injective object.

If is any function, then we have (where "∘" denotes function composition). In particular, is the identity element of the monoid of all functions from to . Since the identity element of a monoid is unique, one can alternately define the identity function on to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of need not be functions.

In other words, is squarefree if and only if h(w) is squarefree for all squarefree of length 3. It is possible to find a squarefree morphism by brute-force search. algorithm squarefree_morphism is output: a squarefree morphism with the lowest possible rank . set k = 3 while True do set k_sf_words to the list of all squarefree words of length over a ternary alphabet for each h(0) in k_sf_words do for each h(1) in k_sf_words do for each h(2) in k_sf_words do if h(1) = h(2) then break from the current loop (advance to next h(1)) if h(0) e h(1) and h(2) e h(0) then if h(w) is squarefree for all squarefree of length then return h(0), h(1), h(2) increment by Over a ternary alphabet, there are exactly 144 uniform squarefree morphisms of rank 11 and no uniform squarefree morphisms with a lower rank than 11.

In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p.. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater... The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. The question addresses the existence of a Galois extension L of K(C), with G as Galois group, and with specified ramification. From a geometric point of view, L corresponds to another curve C′, together with a morphism :π : C′ → C. Geometrically, the assertion that π is ramified at a finite set S of points on C means that π restricted to the complement of S in C is an étale morphism. This is in analogy with the case of Riemann surfaces.

The face map di drops the i-th element from such a list, and the degeneracy maps si duplicates the i-th element. A similar construction can be performed for every category C, to obtain the nerve NC of C. Here, NC([n]) is the set of all functors from [n] to C, where we consider [n] as a category with objects 0,1,...,n and a single morphism from i to j whenever i ≤ j. Concretely, the n-simplices of the nerve NC can be thought of as sequences of n composable morphisms in C: a0 -> a1 -> ... -> an. (In particular, the 0-simplices are the objects of C and the 1-simplices are the morphisms of C.) The face map d0 drops the first morphism from such a list, the face map dn drops the last, and the face map di for 0 < i < n drops ai and composes the ith and (i + 1)th morphisms.

An associative algebra over K is given by a K-vector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A × A → A is reinterpreted as a linear map (i. e., morphism in the category of K-vector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.

If the elements of B are viewed as "propositions about some object", then a 2-valued morphism on B can be interpreted as representing a particular "state of that object", namely the one where the propositions of B which are mapped to 1 are true, and the propositions mapped to 0 are false. Since the morphism conserves the Boolean operators (negation, conjunction, etc.), the set of true propositions will not be inconsistent but will correspond to a particular maximal conjunction of propositions, denoting the (atomic) state. (The true propositions form an ultrafilter, the false propositions form a maximal ideal, as mentioned above.) The transition between two states s1 and s2 of B, represented by 2-valued morphisms, can then be represented by an automorphism f from B to B, such that s2 o f = s1. The possible states of different objects defined in this way can be conceived as representing potential events.

This is an initial morphism from Y to G, because any set map from Y to the underlying set GW of some group W will factor through \eta_Y:Y\to GFY via a unique group homomorphism from FY to W. This is precisely the universal property of the free group on Y. Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let \varepsilon_X:FGX\to X be the group homomorphism which sends the generators of FGX to the elements of X they correspond to, which exists by the universal property of free groups. Then each (GX,\varepsilon_X) is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through \varepsilon_X:FGX\to X via a unique set map from Z to GX. This means that (F,G) is an adjoint pair. Hom-set adjunction.

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.

In homological algebra and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper Géometrie Algébrique et Géométrie Analytique. See also flat morphism.

For this reason, in constructive topology especially, the apartness relation over a set is often taken as primitive, and equality is a defined relation. A set endowed with an apartness relation is known as a constructive setoid. A function f: A \rarr B where A and B are constructive setoids is called a morphism for #A and #B if \forall x, \, y: A.\, f(x) \; \\#_B \; f(y) \Rarr x \; \\#_A \; y.

The above algebraic definition of a semilattice suggests a notion of morphism between two semilattices. Given two join-semilattices and , a homomorphism of (join-) semilattices is a function f: S → T such that :f(x ∨ y) = f(x) ∨ f(y). Hence f is just a homomorphism of the two semigroups associated with each semilattice. If S and T both include a least element 0, then f should also be a monoid homomorphism, i.e.

In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes . The basic insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold.

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.

In fact, this is necessary and sufficient, because if is any nilpotent, then one of its powers will be nilpotent of order at most . In particular, if is a field then the Frobenius endomorphism is injective. The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in .

The structure maps of the operad (the composition and the actions of the symmetric groups) must then be assumed to be continuous. The result is called a topological operad. Similarly, in the definition of a morphism, it would be necessary to assume that the maps involved are continuous. Other common settings to define operads include, for example, module over a ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc.

Chu spaces can serve as a model of concurrent computation in automata theory to express branching time and true concurrency. Chu spaces exhibit the quantum mechanical phenomena of complementarity and uncertainty. The complementarity arises as the duality of information and time, automata and schedules, and states and events. Uncertainty arises when a measurement is defined to be a morphism such that increasing structure in the observed object reduces the clarity of observation.

For example, the closed subscheme of the affine line defined by x2 = 0 is different from the subscheme defined by x = 0 (the origin). More generally, the fiber of a morphism of schemes X → Y at a point of Y may be non-reduced, even if X and Y are reduced. Geometrically, this says that fibers of good mappings may have nontrivial "infinitesimal" structure. There are further generalizations called algebraic spaces and stacks.

Given a left G-module M, it can be turned into a right G-module by defining a·g = g−1·a. A function f : M → N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant. The collection of left (respectively right) G-modules and their morphisms form an abelian category G-Mod (resp. Mod-G). The category G-Mod (resp.

The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result: Theorem Let X and Y be two integral locally noetherian schemes and f \colon X \to Y a proper dominant morphism. Denote their function fields by K(X) and K(Y), respectively. Suppose that the algebraic closure of K(Y) in K(X) has separable degree n and that y \in Y is unibranch.

Informally, category theory is a general theory of functions. Several terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself. Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations from 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.

In algebra, epimorphisms are often defined as surjective homomorphisms. On the other hand, in category theory, epimorphisms are defined as right cancelable morphisms. This means that a (homo)morphism f: A \to B is an epimorphism if, for any pair g, h of morphisms from B to any other object C, the equality g \circ f = h \circ f implies g = h. A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures.

A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow X --\to Y, is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions.

As a graded Hopf algebra, the dual of the ring of quasisymmetric functions is the ring of noncommutative symmetric functions. Every symmetric function is also a quasisymmetric function, and hence the ring of symmetric functions is a subalgebra of the ring of quasisymmetric functions. The ring of quasisymmetric functions is the terminal object in category of graded Hopf algebras with a single character. Hence any such Hopf algebra has a morphism to the ring of quasisymmetric functions.

There are general conjectures due to Shouwu Zhang and others concerning subvarieties that contain infinitely many periodic points or that intersect an orbit in infinitely many points. These are dynamical analogues of, respectively, the Manin-Mumford conjecture, proven by Raynaud, and the Mordell–Lang conjecture, proven by Faltings. The following conjectures illustrate the general theory in the case that the subvariety is a curve. :Conjecture. Let be a morphism and let be an irreducible algebraic curve.

Pic. 9: Monotonic map f between lattices that preserves neither joins nor meets, since and . The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices and , a lattice homomorphism from L to M is a function such that for all : : f(a ∨L b) = f(a) ∨M f(b), and : f(a ∧L b) = f(a) ∧M f(b). Thus f is a homomorphism of the two underlying semilattices.

Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces. In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type.

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].

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 dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here: A co-cone of a diagram F : J → C is an object N of C together with a family of morphisms :\psi_X:F(X) \to N for every object X of J, such that for every morphism f : X → Y in J, we have ψY ∘ F(f) = ψX. A colimit of a diagram F : J → C is a co-cone (L, \varphi) of F such that for any other co- cone (N, ψ) of F there exists a unique morphism u : L → N such that u o \varphiX = ψX for all X in J. A universal co-cone Colimits are also referred to as universal co-cones. They can be characterized as initial objects in the category of co-cones from F. As with limits, if a diagram F has a colimit then this colimit is unique up to a unique isomorphism.

Category theory is a branch of mathematics that formalizes the notion of a special function via arrows or morphisms. A category is an algebraic object that (abstractly) consists of a class of objects, and for every pair of objects, a set of morphisms. A partial (equiv. dependently typed) binary operation called composition is provided on morphisms, every object has one special morphism from it to itself called the identity on that object, and composition and identities are required to obey certain relations.

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 .

In the context of algebraic geometry, the notion of branch points can be generalized to mappings between arbitrary algebraic curves. Let ƒ:X → Y be a morphism of algebraic curves. By pulling back rational functions on Y to rational functions on X, K(X) is a field extension of K(Y). The degree of ƒ is defined to be the degree of this field extension [K(X):K(Y)], and ƒ is said to be finite if the degree is finite.

Complete group varieties are called abelian varieties. This generalizes to the notion of abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers They are automatically projective, and they have many applications, e.g., in geometric class field theory and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff.

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,Mac Lane, p. 126 and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements.

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.

Isogenous elliptic curves to E can be obtained by quotienting E by finite subgroups, here subgroups of the 4-torsion subgroup. For abelian varieties, such as elliptic curves, this notion can also be formulated as follows: Let E1 and E2 be abelian varieties of the same dimension over a field k. An isogeny between E1 and E2 is a dense morphism f : E1 → E2 of varieties that preserves basepoints (i.e. f maps the identity point on E1 to that on E2).

Whereas Darwin spent just five weeks in the Galápagos, and David Lack spent three months, Peter and Rosemary Grant and their colleagues have made research trips to the Galápagos for about thirty years, particularly studying Darwin's finches. The Española cactus finch (Geospiza conirostris) lives on Isla Genovesa (formerly Tower Island) which is formed from a shield volcano, and is home to a variety of birds. These birds, like all well-studied groups,Huxley, Julian S. 1954 (presentation; printed 1955). "Morphism in Birds".

If C and D are two categories and F and G are two functors from C to D, the inserter category Ins(F, G) is the category whose objects are pairs (X, f) where X is an object of C and f is a morphism in D from F(X) to G(X) and whose morphisms from (X, f) to (Y, g) are morphisms h in C from X to Y such that G(h) \circ f = g \circ F(h).

Much of the literature focuses on strict 2-groups. A strict 2-group is a strict monoidal category in which every morphism is invertible and every object has a strict inverse (so that xy and yx are actually equal to the unit object). A strict 2-group is a group object in a category of categories; as such, they are also called groupal categories. Conversely, a strict 2-group is a category object in the category of groups; as such, they are also called categorical groups.

If both F and G are contravariant, the vertical arrows in this diagram are reversed. If \eta is a natural transformation from F to G , we also write \eta : F \to G or \eta : F \implies G . This is also expressed by saying the family of morphisms \eta_X: F(X) \to G(X) is natural in X . If, for every object X in C , the morphism \eta_X is an isomorphism in D , then \eta is said to be a ' (or sometimes natural equivalence' or isomorphism of functors).

Given a morphism of vector bundles E, F of ranks m and n over a smooth variety, its k-th degeneracy locus (k ≤ min(m,n)) is the variety of points where it has rank at most k. If all components of the degeneracy locus have the expected codimension (m – k)(n – k) then Porteous's formula states that its fundamental class is the determinant of the matrix of size m – k whose (i, j) entry is the Chern class cn–k+j–i(F – E).

There is a categorical picture of paths which is sometimes useful. Any topological space X gives rise to a category where the objects are the points of X and the morphisms are the homotopy classes of paths. Since any morphism in this category is an isomorphism this category is a groupoid, called the fundamental groupoid of X. Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point x0 in X is just the fundamental group based at x0.

The degeneracy maps si lengthen the sequence by inserting an identity morphism at position i. We can recover the poset S from the nerve NS and the category C from the nerve NC; in this sense simplicial sets generalize posets and categories. Another important class of examples of simplicial sets is given by the singular set SY of a topological space Y. Here SYn consists of all the continuous maps from the standard topological n-simplex to Y. The singular set is further explained below.

This is equivalent to the above notion, as every dense morphism between two abelian varieties of the same dimension is automatically surjective with finite fibres, and if it preserves identities then it is a homomorphism of groups. Two abelian varieties E1 and E2 are called isogenous if there is an isogeny E1 → E2. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.

Ruled surfaces of genus g have a smooth morphism to a curve of genus g whose fibers are lines P1. They are all algebraic. (The ones of genus 0 are the Hirzebruch surfaces and are rational.) Any ruled surface is birationally equivalent to P1 × C for a unique curve C, so the classification of ruled surfaces up to birational equivalence is essentially the same as the classification of curves. A ruled surface not isomorphic to P1 × P1 has a unique ruling (P1 × P1 has two).

Adalia bipunctata, the two-spotted ladybird, is highly polymorphic. Its basic form is red with two black spots, but it has many other forms, the most important being melanic, with black elytra and red spots. The curious fact about this morphism is that, although the melanic forms are more common in industrial areas, its maintenance has nothing to do with cryptic camouflage and predation. The Coccinellidae as a whole are highly noxious, and experiments with birds and other predators have found this species quite exceptionally distasteful.

Multisets appeared explicitly in the work of Richard Dedekind. Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example, Whitney (1933) described generalized sets ("sets" whose characteristic functions may take any integer value - positive, negative or zero). Monro (1987) investigated the category Mul of multisets and their morphisms, defining a multiset as a set with an equivalence relation between elements "of the same sort", and a morphism between multisets as a function which respects sorts.

Such a (projective) completion always exists and is unique. If the base field is not perfect, a smooth completion of a smooth affine curve doesn't always exist. But the above process always produces a regular completion if we start with a regular affine curve (smooth varieties are regular, and the converse is true over perfect fields). A regular completion is unique and, by the valuative criterion of properness, any morphism from the affine curve to a complete algebraic variety extends uniquely to the regular completion.

This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category. In Ab, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism f : A → B is the subgroup K of A defined by K = {x ∈ A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for cokernels; the cokernel of f is the quotient group C = B / f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup of B, and that therefore the quotient group B / f(A) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category. The product in Ab is given by the product of groups, formed by taking the cartesian product of the underlying sets and performing the group operation componentwise.

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 EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of Zariski : :If f:X->Y is a quasi-projective morphism of Noetherian schemes then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y. In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of quasi-finite morphisms, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that open immersions and finite morphisms are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such : :if Y is a quasi-compact separated scheme and f: X \to Y is a separated, quasi-finite, finitely presented morphism then there is a factorization into X \to Z \to Y, where the first map is an open immersion and the second one is finite. The relation between this theorem about quasi-finite morphisms and Théorème 4.4.

Monoids can be viewed as a special class of categories. Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object. That is, : A monoid is, essentially, the same thing as a category with a single object. More precisely, given a monoid , one can construct a small category with only one object and whose morphisms are the elements of M. The composition of morphisms is given by the monoid operation •.

For example if P is a polynomial of degree 6 (without repeated roots) then :y2 = P(x) is an affine curve representation of a genus 2 curve, necessarily hyperelliptic, and a basis of the differentials of the first kind is given in the same notation by :dx/, x dx/. This means that the canonical map is given by homogeneous coordinates [1: x] as a morphism to the projective line. The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in x.

In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint- free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi- ampleness is a kind of "nonnegativity". More strongly, a line bundle on X is very ample if it has enough sections to give a closed immersion (or "embedding") of X into projective space.

In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism. Normal varieties were introduced by .

In fact, the parallel of f is an isomorphism for every morphism f if and only if the pre-abelian category is an abelian category. An example of a non-abelian, pre-abelian category is, once again, the category of topological abelian groups. As remarked, the image is the inclusion of the closure of the range; however, the coimage is a quotient map onto the range itself. Thus, the parallel is the inclusion of the range into its closure, which is not an isomorphism unless the range was already closed.

Given two reductive groups and a (well behaved) morphism between their corresponding L-groups, this conjecture relates their automorphic representations in a way that is compatible with their L-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an induced representation construction—what in the more traditional theory of automorphic forms had been called a 'lifting', known in special cases, and so is covariant (whereas a restricted representation is contravariant). Attempts to specify a direct construction have only produced some conditional results.

An endomorphism is a homomorphism whose domain equals the codomain, or, more generally, a morphism whose source is equal to the target. The endomorphisms of an algebraic structure, or of an object of a category form a monoid under composition. The endomorphisms of a vector space or of a module form a ring. In the case of a vector space or a free module of finite dimension, the choice of a basis induces a ring isomorphism between the ring of endomorphisms and the ring of square matrices of the same dimension.

An isogeny from an abelian variety A to another one B is a surjective morphism with finite kernel. Some theorems on abelian varieties require the idea of abelian variety up to isogeny for their convenient statement. For example, given an abelian subvariety A1 of A, there is another subvariety A2 of A such that :A1 × A2 is isogenous to A (Poincaré's reducibility theorem: see for example Abelian Varieties by David Mumford). To call this a direct sum decomposition, we should work in the category of abelian varieties up to isogeny.

In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent. Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme.

Let E and D be elliptic curves over a field k. An isogeny between E and D is a finite morphism f : E → D of varieties that preserves basepoints (in other words, maps the given point on E to that on D). The two curves are called isogenous if there is an isogeny between them. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. Every isogeny is an algebraic homomorphism and thus induces homomorphisms of the groups of the elliptic curves for k-valued points.

Every partially ordered set can be viewed as a category in a natural way: there is a unique morphism from x to y if and only if . A monotone Galois connection is then nothing but a pair of adjoint functors between two categories that arise from partially ordered sets. In this context, the upper adjoint is the right adjoint while the lower adjoint is the left adjoint. However, this terminology is avoided for Galois connections, since there was a time when posets were transformed into categories in a dual fashion, i.e.

A famous puzzle in human genetics is the genetic ability to taste phenylthiocarbamide (phenylthiourea or PTC), a morphism which was discovered in 1931. This substance, which is bitter to some people and tasteless to others, is of no great significance in itself, yet it is a genetic dimorphism. Because of its high frequency (which varies in different ethnic groups) it must be connected to some function of selective value. The ability to taste PTC itself is correlated with the ability to taste other bitter substances, many of which are toxic.

The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves. The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem. Conversely, complex analytic analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published.

Except for notational trouble, it is easy to say what such equations are: they express two ways of factoring products of coordinates from the tensor product, obtained in two different ways as something from U times something from V. This mapping or morphism σ is the Segre embedding. Counting dimensions, it shows how the product of projective spaces of dimensions m and n embeds in dimension :(m + 1)(n + 1) - 1 = mn + m + n.\ Classical terminology calls the coordinates on the product multihomogeneous, and the product generalised to k factors k-way projective space.

If M is a commutative monoid, it is acted on naturally by the monoid N of positive integers under multiplication, with an element n of N multiplying an element of M by n. The Frobenioid of M is the semidirect product of M and N. The underlying category of this Frobenioid is category of the monoid, with one object and a morphism for each element of the monoid. The standard Frobenioid is the special case of this construction when M is the additive monoid of non-negative integers.

In category theory, a branch of mathematics, a subterminal object is an object X of a category C with the property that every object of C has at most one morphism into X. If X is subterminal, then the pair of identity morphisms (1X, 1X) makes X into the product of X and X. If C has a terminal object 1, then an object X is subterminal if and only if it is a subobject of 1, hence the name. The category of categories with subterminal objects and functors preserving them is not accessible.

This is not a group because two operations A and B can only be composed if the empty point after carrying out A is the empty point at the beginning of B. It is in fact a groupoid (a category such that every morphism is invertible) whose 13 objects are the 13 points, and whose morphisms from x to y are the operations taking the empty point from x to y. The morphisms fixing the empty point form a group isomorphic to the Mathieu group M12 with 12×11×10×9×8 elements.

Gauss started to write an eighth section on higher-order congruences, but did not complete it, and it was published separately after his death as a treatise titled "general investigations on congruences". In it Gauss discussed congruences of arbitrary degree, attacking the problem of general congruences from a standpoint closely related to that taken later by Dedekind, Galois, and Emil Artin. The treatise paved the way for the theory of function fields over a finite field of constants. Ideas unique to that treatise are clear recognition of the importance of the Frobenius morphism, and a version of Hensel's lemma.

Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G. An infranatural transformation \eta from F to G is simply a family of morphisms \eta_X : F(X) \to G(X) , for all X in C. Thus a natural transformation is an infranatural transformation for which \eta_Y \circ F(f) = G(f) \circ \eta_X for every morphism f : X \to Y . The naturalizer of \eta , nat (\eta) , is the largest subcategory of C containing all the objects of C on which \eta restricts to a natural transformation.

This leads to the idea of using multiple groupoid objects in homotopy theory. More generally, the Eckmann–Hilton argument is a special case of the use of the interchange law in the theory of (strict) double and multiple categories. A (strict) double category is a set, or class, equipped with two category structures, each of which is a morphism for the other structure. If the compositions in the two category structures are written \circ, \otimes then the interchange law reads : (a \circ b) \otimes (c \circ d) = (a \otimes c) \circ (b \otimes d) whenever both sides are defined.

In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is an expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern classes. Giambelli's formula is roughly the special case when the vector bundles are sums of line bundles over projective space. pointed out that the fundamental class must be a polynomial in the Chern classes and found this polynomial in a few special cases, and found the polynomial in general. proved a more general version, and generalized it further.

In algebraic geometry, the Kempf vanishing theorem, introduced by , states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic. and found simpler proofs of the Kempf vanishing theorem using the Frobenius morphism.

In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change. The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context).

In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i\colon A\hookrightarrow X. Sometimes i is assumed to be a cofibration. A morphism from (X,A) to (X',A') is given by two maps f\colon X\rightarrow X' and g\colon A\rightarrow A' such that i' \circ g =f \circ i . A pair of spaces is an ordered pair where is a topological space and a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of by .

In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism : is a closed map, i.e. maps closed sets onto closed sets.Here the product variety X × Y does not carry the product topology, in general; the Zariski topology on it will have more closed sets (except in very simple cases). This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products.

In the special case that the original map was a fibration with fiber , then the homotopy equivalence given above will be a map of fibrations over . This will induce a morphism of their long exact sequences of homotopy groups, from which (by applying the Five Lemma, as is done in the Puppe sequence) one can see that the map is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one. The homotopy fiber is dual to the mapping cone, much as the mapping path space is dual to the mapping cylinder.

Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any surface. The theorem states that any nontrivial birational morphism f\colon X\to Y must contract a −1-curve to a smooth point, and conversely any such curve can be smoothly contracted. Here a −1-curve is a smooth rational curve C with self-intersection C\cdot C = -1.

Eval and apply are the two interdependent components of the eval-apply cycle, which is the essence of evaluating Lisp, described in SICP.The Metacircular Evaluator (SICP Section 4.1) In category theory, the eval morphism is used to define the closed monoidal category. Thus, for example, the category of sets, with functions taken as morphisms, and the cartesian product taken as the product, forms a Cartesian closed category. Here, eval (or, properly speaking, apply) together with its right adjoint, currying, form the simply typed lambda calculus, which can be interpreted to be the morphisms of Cartesian closed categories.

In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.

If ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~. A frequent particular case occurs when f is a function from X to another set Y; if x1 ~ x2 implies f(x1) = f(x2) then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in the character theory of finite groups.

Put another way, there are coverings in that setting that cannot be 'seen' by Galois theory. (See radical morphism for a higher-level discussion.) If L is the field extension :K(T1/p), in other words the splitting field of P, then L/K is an example of a purely inseparable field extension. It is of degree p, but has no automorphism fixing K, other than the identity, because T1/p is the unique root of P. This shows directly that Galois theory must here break down. A field such that there are no such extensions is called perfect.

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).

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.

There are for general reasons free resolutions of R as graded module over K[X0, X1, X2, ..., XN]. A resolution is defined as minimal if the image in each module morphism of free modules :φ:Fi → Fi − 1 in the resolution lies in JFi − 1, where J is the irrelevant ideal. As a consequence of Nakayama's lemma, φ then takes a given basis in Fi to a minimal set of generators in Fi − 1. The concept of minimal free resolution is well-defined in a strong sense: unique up to isomorphism of chain complexes and occurring as a direct summand in any free resolution.

Diagram chasing (also called diagrammatic search) is a method of mathematical proof used especially in homological algebra, where one establishes a property of some morphism by tracing the elements of a commutative diagram. A proof by diagram chasing typically involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences. A syllogism is constructed, for which the graphical display of the diagram is just a visual aid. It follows that one ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified.

DFAO generating the Thue–Morse sequence The Thue–Morse sequence t(n) () is the fixed point of the morphism 0 → 01, 1 → 10. Since the n-th term of the Thue–Morse sequence counts the number of ones modulo 2 in the base-2 representation of n, it is generated by the two-state deterministic finite automaton with output pictured here, where being in state q0 indicates there are an even number of ones in the representation of n and being in state q1 indicates there are an odd number of ones. Hence, the Thue–Morse sequence is 2-automatic.

The five lemma is often applied to long exact sequences: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine the unknown homology groups, but if one can compare the original object and sub object to well- understood ones via morphisms, then a morphism between the respective long exact sequences is induced, and the five lemma can then be used to determine the unknown homology groups.

An action of a linear algebraic group G on a variety (or scheme) X over a field k is a morphism :G \times_k X \to X that satisfies the axioms of a group action. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects. Part of the theory of group actions is geometric invariant theory, which aims to construct a quotient variety X/G, describing the set of orbits of a linear algebraic group G on X as an algebraic variety. Various complications arise.

The category of OX- modules over a fixed locally ringed space (X, OX) is an abelian category. An important subcategory of the category of OX-modules is the category of quasi- coherent sheaves on X. A sheaf of OX-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free OX-modules. A coherent sheaf F is a quasi-coherent sheaf which is, locally, of finite type and for every open subset U of X the kernel of any morphism from a free OU- modules of finite rank to FU is also of finite type.

The system E is satisfiable in S if there is a map f from X to S, which extends to a semigroup morphism f from X+ to S, such that for all (u,v) in E we have f(u) = f(v) in S. Such an f is a solution, or satisfying assignment, for the system E.Lothaire (2011) p. 444 Two systems of equations are equivalent if they have the same set of satisfying assignments. A system of equations if independent if it is not equivalent to a proper subset of itself. A semigroup is compact if every independent system of equations is finite.

That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel. Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions. For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function. For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary set-theoretic concept.

A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension); Here are some properties of linear mappings \Lambda: X \to Y whose proofs are so easy that we omit them; it is assumed that A \subset X and B \subset Y: for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory, it is a morphism in the category of modules over a given ring.

Let S be a Dedekind scheme, X any connected scheme (not necessarily reduced)M. Antei, The fundamental group scheme of a non reduced scheme, Bulletin des Sciences Mathématiques, Volume 135, Issue 5, July–August 2011, Pages 531-539. and X\to S a faithfully flat morphism of finite type (not necessarily proper). Assume the existence of a section x:S\to X. Once we prove that the category of isomorphism classes of torsors over X (pointed over x) under the action of finite and flat S-group schemes is cofiltered then we define the universal torsor (pointed over x) as the projective limit of all the torsors of that category.

It was devised by Voevodsky a few months after the 1996 preprint appeared. Implementing this scheme required making substantial advances in the field of motivic homotopy theory as well as finding a way to build algebraic varieties with a specified list of properties. From the motivic homotopy theory the proof required the following: #A construction of the motivic analog of the basic ingredient of the Spanier–Whitehead duality in the form of the motivic fundamental class as a morphism from the motivic sphere to the Thom space of the motivic normal bundle over a smooth projective algebraic variety. #A construction of the motivic analog of the Steenrod algebra.

One may generalize the notion of the heap of a group to the case of a groupoid which has two objects A and B when viewed as a category. The elements of the heap may be identified with the morphisms from A to B, such that three morphisms x, y, z define a heap operation according to: :[x,y,z] = x y^{-1} z . This reduces to the heap of a group if a particular morphism between the two objects is chosen as the identity. This intuitively relates the description of isomorphisms between two objects as a heap and the description of isomorphisms between multiple objects as a groupoid.

More generally, let be a morphism of degree at least two defined over a number field . Northcott's theorem says that has only finitely many preperiodic points in , and the general Uniform Boundedness Conjecture says that the number of preperiodic points in may be bounded solely in terms of , the degree of , and the degree of over . The Uniform Boundedness Conjecture is not known even for quadratic polynomials over the rational numbers . It is known in this case that cannot have periodic points of period four, five, or six, although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer.

In algebraic geometry, a morphism of schemes :f: X -> Y is called radicial or universally injective, if, for every field K the induced map X(K) → Y(K) is injective. (EGA I, (3.5.4)) This is a generalization of the notion of a purely inseparable extension of fields (sometimes called a radicial extension, which should not be confused with a radical extension.) It suffices to check this for K algebraically closed. This is equivalent to the following condition: f is injective on the topological spaces and for every point x in X, the extension of the residue fields :k(f(x)) ⊂ k(x) is radicial, i.e.

For some categories C the direct limit used to define the stalk may not exist. However, it exists for most categories which occur in practice, such as the category of sets or most categories of algebraic objects such as abelian groups or rings, which are namely cocomplete. There is a natural morphism F(U) → Fx for any open set U containing x: it takes a section s in F(U) to its germ, that is, its equivalence class in the direct limit. This is a generalization of the usual concept of a germ, which can be recovered by looking at the stalks of the sheaf of continuous functions on X.

Consider the group (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of is its second factor . Note that the first factor, , contains subgroups isomorphic to , for instance ; let be the morphism mapping onto the indicated subgroup. Then the composition of the projection of onto its second factor , followed by , followed by the inclusion of into as its first factor, provides an endomorphism of under which the image of the center, , is not contained in the center, so here the center is not a fully characteristic subgroup of .

Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. In other words, it is a ringed space which is locally a spectrum of a commutative ring. The relative point of view is that much of algebraic geometry should be developed for a morphism X → Y of schemes (called a scheme X over Y), rather than for an individual scheme.

It is equivalent to require that around each x, there exists an open affine subset such that , where f is a non-zero divisor in A. The sum of two effective Cartier divisors corresponds to multiplication of ideal sheaves. There is a good theory of families of effective Cartier divisors. Let be a morphism. A relative effective Cartier divisor for X over S is an effective Cartier divisor D on X which is flat over S. Because of the flatness assumption, for every S'\to S, there is a pullback of D to X \times_S S', and this pullback is an effective Cartier divisor.

Let , and be morphisms of a category containing objects A, B, C and D. By repeated composition, we can construct a morphism from A to D in two ways: :, and :. We have now the following coherence statement: :. In these two particular examples, the coherence statements are theorems for the case of an abstract category, since they follow directly from the axioms; in fact, they are axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.

Such a pair is called a Chu transform or morphism of Chu spaces. A topological space (X, T) where X is the set of points and T the set of open sets, can be understood as a Chu space (X,∈,T) over {0, 1}. That is, the points of the topological space become those of the Chu space while the open sets become states and the membership relation " ∈ " between points and open sets is made explicit in the Chu space. The condition that the set of open sets be closed under arbitrary (including empty) union and finite (including empty) intersection becomes the corresponding condition on the columns of the matrix.

In the context of schemes, the importance of ideal sheaves lies mainly in the correspondence between closed subschemes and quasi-coherent ideal sheaves. Consider a scheme X and a quasi-coherent ideal sheaf J in OX. Then, the support Z of OX/J is a closed subspace of X, and (Z, OX/J) is a scheme (both assertions can be checked locally). It is called the closed subscheme of X defined by J. Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map : i#: OX → i⋆OZ is surjective on the stalks.

Then, the kernel J of i# is a quasi-coherent ideal sheaf, and i induces an isomorphism from Z onto the closed subscheme defined by J.EGA I, 4.2.2 b) A particular case of this correspondence is the unique reduced subscheme Xred of X having the same underlying space, which is defined by the nilradical of OX (defined stalk-wise, or on open affine charts).EGA I, 5.1 For a morphism f: X → Y and a closed subscheme Y′ ⊆ Y defined by an ideal sheaf J, the preimage Y′ ×Y X is defined by the ideal sheafEGA I, 4.4.5 : f⋆(J)OX = im(f⋆J → OX).

Similarly, the projection X → D is a degree 2 morphism ramified over the contact points on D of the four lines tangent to both C and D, and the corresponding involution \tau has the form x → q − x for some q. Thus the composition \tau \sigma is a translation on X. If a power of \tau \sigma has a fixed point, that power must be the identity. Translated back into the language of C and D, this means that if one point c ∈ C (equipped with a corresponding d) gives rise to an orbit that closes up (i.e., gives an n-gon), then so does every point.

A resolution of singularities :f:X\to Y of a complex variety Y is called a small resolution if for every r > 0, the space of points of Y where the fiber has dimension r is of codimension greater than 2r. Roughly speaking, this means that most fibers are small. In this case the morphism induces an isomorphism from the (intersection) homology of X to the intersection homology of Y (with the middle perversity). There is a variety with two different small resolutions that have different ring structures on their cohomology, showing that there is in general no natural ring structure on intersection (co)homology.

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.

A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component f\colon V\to W is a (quasi) isomorphism, where the differentials of V and W are just the linear components of m_V and m_W. An important special class of homotopy Lie algebras are the so-called minimal homotopy Lie algebras, which are characterized by the vanishing of their linear component l_1. This means that any quasi isomorphism of minimal homotopy Lie algebras must be an isomorphism. Any homotopy Lie algebra is quasi-isomorphic to a minimal one, which must be unique up to isomorphism and it is therefore called its minimal model.

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.

One of the key properties of A_\infty-algebras is their structure can be transferred to other algebraic objects given the correct hypotheses. An early rendition of this property was the following: Given an A_\infty-algebra A^\bullet and a homotopy equivalence of complexes > f:B^\bullet \to A^\bullet then there is an A_\infty-algebra structure on B^\bullet inherited from A^\bullet and f can be extended to a morphism of A_\infty-algebras. There are multiple theorems of this flavor with different hypotheses on B^\bullet and f, some of which have stronger results, such as uniqueness up to homotopy for the structure on B^\bullet and strictness on the map f.

In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of X is the category O(Xét) whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology. The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology.

In particular, it forms a category; the objects are taken to be the points of and the collection of morphisms from to is the collection of equivalence classes given above. The fact that this satisfies the definition of a category amounts to the standard fact that the equivalence class of the concatenation of two paths only depends on the equivalence classes of the individual paths.Spanier, section 1.7; Lemma 6 and Theorem 7. Likewise, the fact that this category is a groupoid, which asserts that every morphism is invertible, amounts to the standard fact that one can reverse the orientation of a path, and the equivalence class of the resulting concatenation contains the constant path.

Specifically, let be a finitely presented morphism of pointed schemes and M be an OX-module of finite type whose fiber at x is non- zero. Set n equal to the dimension of and r to the codepth of M at s, that is, to .EGA 0IV, Définition 16.4.9 Then there exist affine étale neighborhoods X′ of x and S′ of s, together with points x′ and s′ lifting x and s, such that the residue field extensions and are trivial, the map factors through S′, this factorization sends x′ to s′, and that the pullback of M to X′ admits a total S′-dévissage at x′ in dimensions between n and .

The fixed points of τ correspond to the boundary points of Σ/τ. The surface Σ is called an "analytic double" of Σ/τ. The Klein surfaces form a category; a morphism from the Klein surface X to the Klein surface Y is a differentiable map f:X→Y which on each coordinate patch is either holomorphic or the complex conjugate of a holomorphic map and furthermore maps the boundary of X to the boundary of Y. There is a one-to-one correspondence between smooth projective algebraic curves over the reals (up to isomorphism) and compact connected Klein surfaces (up to equivalence). The real points of the curve correspond to the boundary points of the Klein surface.

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).

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.

An Algebraic Theory T is a category whose objects are natural numbers 0, 1, 2,..., and which, for each n, has an n-tuple of morphisms: proji: n → 1, i = 1,..., n This allows interpreting n as a cartesian product of n copies of 1. Example. Let's define an algebraic theory T taking hom(n, m) to be m-tuples of polynomials of n free variables X1,..., Xn with integer coefficients and with substitution as composition. In this case proji is the same as Xi. This theory T is called the theory of commutative rings. In an algebraic theory, any morphism n → m can be described as m morphisms of signature n → 1.

In algebraic geometry, Nagata's compactification theorem, introduced by , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper mapping. Nagata's original proof used the older terminology of Zariski–Riemann spaces and valuation theory, which sometimes made it hard to follow. Deligne showed, in unpublished notes expounded by Conrad, that Nagata's proof can be translated into scheme theory and that the condition that S is Noetherian can be replaced by the much weaker condition that S is quasi-compact and quasi- separated. gave another scheme-theoretic proof of Nagata's theorem.

The truth predicate of interest in a typical correspondence theory of truth tells of a relation between representations and objective states of affairs, and is therefore expressed, for the most part, by a dyadic predicate. In general terms, one says that a representation is true of an objective situation, more briefly, that a sign is true of an object. The nature of the correspondence may vary from theory to theory in this family. The correspondence can be fairly arbitrary or it can take on the character of an analogy, an icon, or a morphism, whereby a representation is rendered true of its object by the existence of corresponding elements and a similar structure.

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.

A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist. Let X and Y be objects of a category D with finite products. The product of X and Y is an object X × Y together with two morphisms :\pi_1 : X \times Y \to X :\pi_2 : X \times Y \to Y such that for any other object Z of D and morphisms f: Z \to X and g: Z \to Y there exists a unique morphism h: Z \to X \times Y such that f = \pi_1 \circ h and g = \pi_2 \circ h.

Any affine variety can be completed, in a unique way, into a projective variety by adding its points at infinity, which consists of homogenizing the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by saturating with respect to the homogenizing variable. An important property of projective spaces and projective varieties is that the image of a projective variety under a morphism of algebraic varieties is closed for Zariski topology (that is, it is an algebraic set). This is a generalization to every ground field of the compactness of the real and complex projective space. A projective space is itself a projective variety, being the set of zeros of the zero polynomial.

Dialectica spaces are a categorical way of constructing models of linear logic. They were introduced by Valeria de Paiva, Martin Hyland's student, in her doctoral thesis, as a way of modeling both linear logic and Gödel's dialectica interpretation--hence the name. Given a category C and a specific object K of C with certain (logical) properties, one can construct the category of Dialectica spaces over C, whose objects are pairs of objects of C, related by a C-morphism into the given object. Morphisms of Dialectica spaces are similar to Chu space morphisms, but instead of an equality condition, they have an inequality condition, which is read as a logical implication, the first object implies the second.

A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k. One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed. A k-variety is called a variety if it is absolutely irreducible, i.e. is not the union of two strictly smaller kalg-algebraic sets.

The easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps. Then nX can be set to any multiple of m that is larger than twice the length of X. But the Morse sequence is uniformly recurrent without being periodic, not even eventually periodic (meaning periodic after some nonperiodic initial segment). We define the Thue–Morse morphism to be the function f from the set of binary sequences to itself by replacing every 0 in a sequence with 01 and every 1 with 10. Then if T is the Thue–Morse sequence, then f(T) is T again; that is, T is a fixed point of f.

Let be a smooth map between (smooth) manifolds M and N, and suppose is a smooth function on N. Then the pullback of f by φ is the smooth function φ∗f on M defined by . Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ−1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.) More generally, if is a smooth map from N to any other manifold A, then is a smooth map from M to A.

A normal complex variety X has the property, when viewed as a stratified space using the classical topology, that every link is connected. Equivalently, every complex point x has arbitrarily small neighborhoods U such that U minus the singular set of X is connected. For example, it follows that the nodal cubic curve X in the figure, defined by x2 = y2(y + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from A1 to X which is not an isomorphism; it sends two points of A1 to the same point in X. Curve y2 = x2(x + 1) More generally, a scheme X is normal if each of its local rings :OX,x is an integrally closed domain.

Let X be a topological space which is the union of two open and path connected subspaces U1, U2. Suppose U1 ∩ U2 is path connected and nonempty, and let x0 be a point in U1 ∩ U2 that will be used as the base of all fundamental groups. The inclusion maps of U1 and U2 into X induce group homomorphisms j_1:\pi_1(U_1,x_0)\to \pi_1(X,x_0) and j_2:\pi_1(U_2,x_0)\to \pi_1(X,x_0). Then X is path connected and j_1 and j_2 form a commutative pushout diagram: :750px the natural morphism k is an isomorphism, that is, the fundamental group of X is the free product of the fundamental groups of U1 and U2 with amalgamation of \pi_1(U_1\cap U_2, x_0). pg.

In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due to Alexander Grothendieck. Generic flatness states that if Y is an integral locally noetherian scheme, is a finite type morphism of schemes, and F is a coherent OX-module, then there is a non-empty open subset U of Y such that the restriction of F to u−1(U) is flat over U.EGA IV2, Théorème 6.9.1 Because Y is integral, U is a dense open subset of Y. This can be applied to deduce a variant of generic flatness which is true when the base is not integral.

He shifted attention from the study of individual varieties to the relative point of view (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems. The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite-dimensional; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a one-point space. In 1956, he applied the same thinking to the Riemann–Roch theorem, which had already recently been generalized to any dimension by Hirzebruch. The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.

In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain X with its image f(X) contained in Y, so that .

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.

On the one hand σ-completeness is too weak to characterize inverse image maps (completeness is required), on the other hand it is too restrictive for a generalization. (Sikorski remarked on using non-σ- complete homomorphisms but included σ-completeness in his axioms for closure algebras.) Later J. Schmid defined a continuous homomorphism or continuous morphism for interior algebras as a Boolean homomorphism f between two interior algebras satisfying f(xC) ≤ f(x)C. This generalizes the forward image map of a continuous map - the image of a closure is contained in the closure of the image. This construction is covariant but not suitable for category theoretic applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections. (C.

Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in J). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph G. If we let J be the free category generated by G, there is a universal diagram F : J → C whose image contains G. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms. Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.

That is, each of these rings is an integral domain R, and every ring S with R ⊆ S ⊆ Frac(R) such that S is finitely generated as an R-module is equal to R. (Here Frac(R) denotes the field of fractions of R.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to X is an isomorphism. An older notion is that a subvariety X of projective space is linearly normal if the linear system giving the embedding is complete. Equivalently, X ⊆ Pn is not the linear projection of an embedding X ⊆ Pn+1 (unless X is contained in a hyperplane Pn). This is the meaning of "normal" in the phrases rational normal curve and rational normal scroll.

One construction of the localization is done by declaring that its objects are the same as those in C, but the morphisms are enhanced by adding a formal inverse for each morphism in W. Under suitable hypotheses on W, the morphisms between two objects X, Y are given by roofs :X \stackrel f \leftarrow X' \rightarrow Y (where X' is an arbitrary object of C and f is in the given class W of morphisms), modulo certain equivalence relations. These relations turn the map going in the "wrong" direction into an inverse of f. This procedure, however, in general yields a proper class of morphisms between X and Y. Typically, the morphisms in a category are only allowed to form a set. Some authors simply ignore such set-theoretic issues.

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field. For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over :Spec(R) (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism :Spec(F) -> Spec(R) gives back A. The Néron model is a smooth group scheme, so we can consider A0, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model.

These include the aforementioned ones, where either L is a free K-module (hence whenever K is a field), or K contains the field of rational numbers. More generally, the PBW theorem as formulated above extends to cases such as where (1) L is a flat K-module, (2) L is torsion-free as an abelian group, (3) L is a direct sum of cyclic modules (or all its localizations at prime ideals of K have this property), or (4) K is a Dedekind domain. See, for example, the 1969 paper by Higgins for these statements. Finally, it is worth noting that, in some of these cases, one also obtains the stronger statement that the canonical morphism S(L) → grU(L) lifts to a K-module isomorphism S(L) → U(L), without taking associated graded.

For example, if X is an affine variety, then one can try to construct X/G as Spec of the ring of invariants O(X)G. However, Masayoshi Nagata showed that the ring of invariants need not be finitely generated as a k-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to Hilbert's 14th problem. In the positive direction, the ring of invariants is finitely generated if G is reductive, by Haboush's theorem, proved in characteristic zero by Hilbert and Nagata. Geometric invariant theory involves further subtleties when a reductive group G acts on a projective variety X. In particular, the theory defines open subsets of "stable" and "semistable" points in X, with the quotient morphism only defined on the set of semistable points.

The usual category theoretical definition is in terms of the property of lifting that carries over from free to projective modules: a module P is projective if and only if for every surjective module homomorphism and every module homomorphism , there exists a module homomorphism such that . (We don't require the lifting homomorphism h to be unique; this is not a universal property.) :120px The advantage of this definition of "projective" is that it can be carried out in categories more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property may also be rephrased as every morphism from P to M factors through every epimorphism to M. Thus, by definition, projective modules are precisely the projective objects in the category of R-modules.

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.

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.

The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection. The most natural and convenient way to express the relation involves the algebraic concept of exact sequences: sequences of objects (in this case groups) and morphisms (in this case group homomorphisms) between them such that the image of one morphism equals the kernel of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are topological manifolds, simplicial complexes, or CW complexes, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability.

In category theory, a global element of an object A from a category is a morphism :h\colon 1 \to A, where is a terminal object of the category.. Roughly speaking, global elements are a generalization of the notion of "elements" from the category of sets, and they can be used to import set- theoretic concepts into category theory. However, unlike a set, an object of a general category need not be determined by its global elements (not even up to isomorphism). For example, the terminal object of the category Grph of graph homomorphisms has one vertex and one edge, a self-loop,. whence the global elements of a graph are its self-loops, conveying no information either about other kinds of edges, or about vertices having no self-loop, or about whether two self-loops share a vertex.

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.

The graph K is needed to attach the pattern being matched to its context: if it is empty, the match can only designate a whole connected component of the graph G. In contrast a graph rewriting rule of the SPO approach is a single morphism in the category of labeled multigraphs and partial mappings that preserve the multigraph structure: r\colon L\rightarrow R. Thus a rewriting step is defined by a single pushout diagram. Practical understanding of this is similar to the DPO approach. The difference is, that there is no interface between the host graph G and the graph G' being the result of the rewriting step. From the practical perspective, the key distinction between DPO and SPO is how they deal with the deletion of nodes with adjacent edges, in particular, how they avoid that such deletions may leave behind "dangling edges".

A homomorphism between two Boolean algebras A and B is a function f : A → B such that for all a, b in A: : f(a ∨ b) = f(a) ∨ f(b), : f(a ∧ b) = f(a) ∧ f(b), : f(0) = 0, : f(1) = 1. It then follows that f(¬a) = ¬f(a) for all a in A. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices. An isomorphism between two Boolean algebras A and B is a homomorphism f : A → B with an inverse homomorphism, that is, a homomorphism g : B → A such that the composition g ∘ f: A → A is the identity function on A, and the composition f ∘ g: B → B is the identity function on B. A homomorphism of Boolean algebras is an isomorphism if and only if it is bijective.

More precisely, let V be an algebraic variety over K (assumptions here are: V is an irreducible set, a quasi-projective variety, and K has characteristic zero). A type I thin set is a subset of V(K) that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than d, the dimension of V. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the K-points of some other d-dimensional algebraic variety V′, that maps essentially onto V as a ramified covering with degree e > 1. Saying this more technically, a thin set of type II is any subset of :φ(V′(K)) where V′ satisfies the same assumptions as V and φ is generically surjective from the geometer's point of view.

The surface S is naturally embedded into the grassmannian of lines G(2,5) of P4. Let U be the restriction to S of the universal rank 2 bundle on G. We have the: Tangent bundle Theorem (Fano, Clemens-Griffiths, Tyurin): The tangent bundle of S is isomorphic to U. This is a quite interesting result because, a priori, there should be no link between these two bundles. It has many powerful applications. By example, one can recover the fact that the cotangent space of S is generated by global sections. This space of global 1-forms can be identified with the space of global sections of the tautological line bundle O(1) restricted to the cubic F and moreover: Torelli- type Theorem : Let g' be the natural morphism from S to the grassmannian G(2,5) defined by the cotangent sheaf of S generated by its 5-dimensional space of global sections.

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).

Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ring R is complete. If it is not, then it can be completed: one can find an essentially unique complete topological ring S that contains R as a dense subring such that the given topology on R equals the subspace topology arising from S. If the starting ring R is metric, the ring S can be constructed as a set of equivalence classes of Cauchy sequences in R, this equivalence relation makes the ring S Hausdorff and using constant sequences (which are Cauchy) one realises a (uniformly) continuous morphism (CM in the sequel) such that, for all CM where is Hausdorff and complete, there exists a unique CM such that f=g\circ c. If R is not metric (as, for instance, the ring of all real-variable rational valued functions i.e.

Let X be a variety of nonnegative Kodaira dimension over a field of characteristic zero, and let B be the canonical model of X, B = Proj R(X, KX); the dimension of B is equal to the Kodaira dimension of X. There is a natural rational map X – → B; any morphism obtained from it by blowing up X and B is called the Iitaka fibration. The minimal model and abundance conjectures would imply that the general fiber of the Iitaka fibration can be arranged to be a Calabi–Yau variety, which in particular has Kodaira dimension zero. Moreover, there is an effective Q-divisor Δ on B (not unique) such that the pair (B, Δ) is klt, KB \+ Δ is ample, and the canonical ring of X is the same as the canonical ring of (B, Δ) in degrees a multiple of some d > 0.O. Fujino and S. Mori, J. Diff. Geom.

These generators and relations define K(X), and they imply that it is the universal way to assign invariants to vector bundles in a way compatible with exact sequences. Grothendieck took the perspective that the Riemann–Roch theorem is a statement about morphisms of varieties, not the varieties themselves. He proved that there is a homomorphism from K(X) to the Chow groups of X coming from the Chern character and Todd class of X. Additionally, he proved that a proper morphism to a smooth variety Y determines a homomorphism called the pushforward. This gives two ways of determining an element in the Chow group of Y from a vector bundle on X: Starting from X, one can first compute the pushforward in K-theory and then apply the Chern character and Todd class of Y, or one can first apply the Chern character and Todd class of X and then compute the pushforward for Chow groups.

The concept of a principal homogeneous space can also be globalized as follows. Let X be a "space" (a scheme/manifold/topological space etc.), and let G be a group over X, i.e., a group object in the category of spaces over X. In this case, a (right, say) G-torsor E on X is a space E (of the same type) over X with a (right) G action such that the morphism :E \times_X G \rightarrow E \times_X E given by :(x,g) \mapsto (x,xg) is an isomorphism in the appropriate category, and such that E is locally trivial on X, in that acquires a section locally on X. Isomorphism classes of torsors in this sense correspond to classes in the cohomology group H1(X,G). When we are in the smooth manifold category, then a G-torsor (for G a Lie group) is then precisely a principal G-bundle as defined above.

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.

A group G is said to be linear if there exists a field K, an integer d and an injective morphism from G to the general linear group GLd(K) (a faithful linear representation of dimension d over K): if needed one can mention the field and dimension by saying that G is linear of degree d over K. Basic instances are groups which are defined as subgroups of a linear group, for example: #The group GLn(K) itself; #The special linear group SLn(K) (the subgroup of matrices with determinant 1); #The group of invertible upper (or lower) triangular matrices #If gi is a collection of elements in GLn(K) indexed by a set I, then the subgroup generated by the gi is a linear group. In the study of Lie groups, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented over the field of complex numbers. (Some authors require that the group be represented as a closed subgroup of the GLn(C).) Books that follow this approach include Hall (2015) and Rossman (2002).

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.

Let D = k[ε] be the ring of dual numbers over a field k and Y a scheme of finite type over k. Given a closed subscheme X of Y, by definition, an embedded first-order infinitesimal deformation of X is a closed subscheme X of Y ×Spec(k) Spec(D) such that the projection X → Spec D is flat and has X as the special fiber. If Y = Spec A and X = Spec(A/I) are affine, then an embedded infinitesimal deformation amounts to an ideal I of A[ε] such that A[ε]/ I is flat over D and the image of I in A = A[ε]/ε is I. In general, given a pointed scheme (S, 0) and a scheme X, a morphism of schemes : X → S is called the deformation of a scheme X if it is flat and the fiber of it over the distinguished point 0 of S is X. Thus, the above notion is a special case when S = Spec D and there is some choice of embedding.

A gerbe on a topological space X is a stack G of groupoids over X which is locally non- empty (each point in X has an open neighbourhood U over which the section category G(U) of the gerbe is not empty) and transitive (for any two objects a and b of G(U) for any open set U, there is an open covering {Vi}i of U such that the restrictions of a and b to each Vi are connected by at least one morphism). A canonical example is the gerbe of principal bundles with a fixed structure group H: the section category over an open set U is the category of principal H-bundles on U with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle X x H over X shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.

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.