Presheaf (category theory)

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In category theory, a branch of mathematics, a V-valued presheaf F on a category C is a functor F\colon C^\mathrm{op}\to\mathbf{V}. Often presheaf is defined to be a Set-valued presheaf. 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 into a category, and is an example of a functor category. It is often written as \widehat{C} = \mathbf{Set}^{C^\mathrm{op}}. A functor into \widehat{C} 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.

Examples[edit]

Properties[edit]

  • When C is a small category, the functor category \widehat{C}=\mathbf{Set}^{C^\mathrm{op}} is cartesian closed.
  • The partially ordered set of subobjects of P form a Heyting algebra, whenever P is an object of \widehat{C}=\mathbf{Set}^{C^\mathrm{op}} for small C.
  • For any morphism f:X\to Y of \widehat{C}, the pullback functor of subobjects f^*:\mathrm{Sub}_{\widehat{C}}(Y)\to\mathrm{Sub}_{\widehat{C}}(X) has a right adjoint, denoted \forall_f, and a left adjoint, \exists_f. These are the universal and existential quantifiers.
  • A locally small category C embeds fully and faithfully into the category \widehat{C} of set-valued presheaves via the Yoneda embedding \mathrm{Y}_c which to every object A of C associates the hom-set C(-,A).
  • The presheaf category \widehat{C} is (up to equivalence of categories) the free colimit completion of the category C.

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