Limit And Colimit Of Presheaves

	Limit And Colimit Of Presheaves In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category ''C'' is a limit or colimit in the functor category \widehat = \mathbf(C^, \mathbf). The category \widehat admits small limits and small colimits. Explicitly, if f: I \to \widehat is a functor from a small category ''I'' and ''U'' is an object in ''C'', then \varinjlim_ f(i) is computed pointwise: :(\varinjlim f(i))(U) = \varinjlim f(i)(U). The same is true for small limits. Concretely this means that, for example, a fiber product exists and is computed pointwise. When ''C'' is small, by the Yoneda 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 (vie ..., one can view ''C'' as the full subcategory of \widehat. If \eta: C \to D is a functor, if f: I \to C is a functor from a small catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Category Theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Presheaf (category Theory) In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on C into a category, and is an example of a functor category. It is often written as \widehat = \mathbf^. A functor into \widehat is sometimes called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, ''A'') for some object ''A'' of C is called a representable presheaf. Some authors refer to a functor F\colon C^\mathrm\to\mathbf as a \mathbf-valued presheaf. Examples * A simplicial set is a Set-valued presheaf on the simplex category C=\Delta. Properties * When C is a small category, the functor category \widehat=\mathbf^ is cartesian closed. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Grothendieck Universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' are both elements of ''U'', then \ is an element of ''U''. # If ''x'' is an element of ''U'', then ''P''(''x''), the power set of ''x'', is also an element of ''U''. # If \_ is a family of elements of ''U'', and if is an element of ''U'', then the union \bigcup_ x_\alpha is an element of ''U''. A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.). Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry. The existence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Limit (category Theory) In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite category. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Colimit (category Theory) In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite category. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Yoneda 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Density Theorem (category Theory) In category theory, a branch of mathematics, the density theorem states that every presheaf of sets is a colimit of representable presheaves in a canonical way. For example, by definition, a simplicial set is a presheaf on the simplex category Δ and a representable simplicial set is exactly of the form \Delta^n = \operatorname(-, (called the standard ''n''-simplex) so the theorem says: for each simplicial set ''X'', :X \simeq \varinjlim \Delta^n where the colim runs over an index category determined by ''X''. Statement Let ''F'' be a presheaf on a category ''C''; i.e., an object of the functor category \widehat = \mathbf(C^\text, \mathbf). For an index category over which a colimit will run, let ''I'' be the category of elements of ''F'': it is the category where # an object is a pair (U, x) consisting of an object ''U'' in ''C'' and an element x \in F(U), # a morphism (U, x) \to (V, y) consists of a morphism u: U \to V in ''C'' such that (Fu)(y) = x. It comes with the forg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Category Theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]