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(-,A) for some object A of C is called a representable presheaf.
Some authors refer to a functor as a -valued presheaf.
- 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 category admits small limits and small colimits.. See limit and colimit of presheaves for further discussion.
- The density theorem states that every presheaf is a colimit of representable presheaves; in fact, is the colimit completion of (see #Universal property below.)
The construction is called the colimit completion of C because of the following universal property:
Proposition — Let C, D be categories and assume D admits small colimits. Then each functor factorizes as
where y is the Yoneda embedding and is a colimit-preserving functor called the Yoneda extension of .
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 (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 .
The proposition yields several corollaries. For example, the proposition implies that the construction is functorial: i.e., each functor determines the functor .
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.)