In category theory, a branch of mathematics, a presheaf on a category C is a functor. 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 and it is called the category of presheaves on C. 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. Some authors refer to a functor as a \mathbf{V}-valued presheaf.

Examples

Properties

Universal property

The construction is called the colimit completion of C because of the following universal property: Proof: Given a presheaf F, by the density theorem, we can write where U_i are objects in C. Then let which exists by assumption. Since is functorial, this determines the functor. Succinctly, is the left Kan extension of \eta along y; hence, the name "Yoneda extension". To see commutes with small colimits, we show is a left-adjoint (to some functor). Define to be the functor given by: for each object M in D and each object U in C, Then, for each object M in D, since by the Yoneda lemma, we have: which is to say is a left-adjoint to. \square The proposition yields several corollaries. For example, the proposition implies that the construction is functorial: i.e., each functor C \to D determines the functor.

Variants

A presheaf of spaces on an ∞-category C is a contravariant functor from C to the ∞-category of spaces (for example, the nerve of the category of CW-complexes.) It is an ∞-category version of a presheaf of sets, as a "set" is replaced by a "space". The notion is used, among other things, in the ∞-category formulation of Yoneda's lemma that says: is fully faithful (here C can be just a simplicial set.) A copresheaf of a category C is a presheaf of Cop. In other words, it is a covariant functor from C to Set.