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Stackification
In algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object. Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks. Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category ''C''; for example, ''C'' can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology. Informal definition Let ''F'' be a category and suppose it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid Scheme
In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Definition A groupoid object in a category ''C'' admitting finite fiber products consists of a pair of objects R, U together with five morphisms :s, t: R \to U, \ e: U \to R, \ m: R \times_ R \to R, \ i: R \to R satisfying the following groupoid axioms # s \circ e = t \circ e = 1_U, \, s \circ m = s \circ p_1, t \circ m = t \circ p_2 where the p_i: R \times_ R \to R are the two projections, # (associativity) m \circ (1_R \times m) = m \circ (m \times 1_R), # (unit) m \circ (e \circ s, 1_R) = m \circ (1_R, e \circ t) = 1_R, # (inverse) i \circ i = 1_R, s \circ i = t, \, t \circ i = s, m \circ (1_R, i) = e \circ s, \, m \circ (i, 1_R) = e \circ t. Examples Group objects A group object is a special case of a groupoid object, where R = U and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheafification
In mathematics, the gluing axiom is introduced to define what a sheaf \mathcal F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor ::(X) \rightarrow C to a category C which initially one takes to be the category of sets. Here (X) is the partial order of open sets of X ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism :U \rightarrow V if U is a subset of V, and none otherwise. As phrased in the sheaf article, there is a certain axiom that F must satisfy, for any open cover of an open set of X. For example, given open sets U and V with union X and intersection W, the required condition is that :(X) is the subset of (U) \times (V) With equal image in (W) In less formal language, a section s of F over X is equally well given by a pair of sections :(s', s'') on U and V respectively, which 'agree' in the sense that s' and s'' have a common image in (W) under the respect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne–Mumford Stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks. If the "étale" is weakened to "smooth", then such a stack is called an algebraic stack (also called an Artin stack, after Michael Artin). An algebraic space is Deligne–Mumford. A key fact about a Deligne–Mumford stack ''F'' is that any ''X'' in F(B), where ''B'' is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme. Examples Affine Stacks Deligne–Mumford stacks are typically constructed by taking the stack quotient of some variety where the stabilizers are finite groups. For example, consider the action of the cyclic group C_n = \langle a \mid a^n =1 \rangle on \mathbb^2 given by a\cdot\colon(x,y) \mapsto (\zeta_n x, \zeta_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology. The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem). Definition There are two common ways to define algebraic spaces: they can be defined as either quotients of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda's Lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Generalities The Yoneda lemma suggests that instead of studying the locally small category \mathcal , one should study the category of all functors of \mathcal into \mathbf (the category of sets with f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient By An Equivalence Relation
In mathematics, given a category ''C'', a quotient of an object ''X'' by an equivalence relation f: R \to X \times X is a coequalizer for the pair of maps :R \ \overset\ X \times X \ \overset\ X,\ \ i = 1,2, where ''R'' is an object in ''C'' and "''f'' is an equivalence relation" means that, for any object ''T'' in ''C'', the image (which is a set) of f: R(T) = \operatorname(T, R) \to X(T) \times X(T) is an equivalence relation; that is, a reflexive, symmetric and transitive relation. The basic case in practice is when ''C'' is the category of all schemes over some scheme ''S''. But the notion is flexible and one can also take ''C'' to be the category of sheaves. Examples *Let ''X'' be a set and consider some equivalence relation on it. Let ''Q'' be the set of all equivalence classes in ''X''. Then the map q: X \to Q that sends an element ''x'' to the equivalence class to which ''x'' belongs is a quotient. *In the above example, ''Q'' is a subset of the power set ''H'' of ''X' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coequalizer
In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition 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: 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 " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
