Presheaf (category theory)

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In category theory, a branch of mathematics, a presheaf on a category is a functor . If 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 . A functor into 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 as a -valued presheaf[citation needed].



  • When is a small category, the functor category is cartesian closed.
  • The partially ordered set of subobjects of form a Heyting algebra, whenever is an object of for small .
  • For any morphism of , the pullback functor of subobjects has a right adjoint, denoted , and a left adjoint, . These are the universal and existential quantifiers.
  • A locally small category embeds fully and faithfully into the category of set-valued presheaves via the Yoneda embedding which to every object of associates the hom functor .
  • The presheaf category is (up to equivalence of categories) the free colimit completion of the category .

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