Presheaf (category theory)


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 on 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 for some object A of C is called a representable presheaf.
Some authors refer to a functor as a -valued presheaf.

Examples

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 are objects in C. Then let which exists by assumption. Since is functorial, this determines the functor. Succinctly, is the left Kan extension of along y; hence, the name "Yoneda extension". To see commutes with small colimits, we show is a left-adjoint. 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.
The proposition yields several corollaries. For example, the proposition implies that the construction is functorial: i.e., each functor determines the functor.

Variants

A presheaf of spaces on an ∞-category C is a contravariant functor from C to the ∞-category of spaces 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