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 and it is called the category of presheaves on . 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.
Examples A simplicial set is a Set-valued presheaf on the simplex category . A directed multigraph is a presheaf on the category with two objects and two parallel morphisms between them i.e. . An arrow category is a presheaf on the category with two objects and one morphism between them. i.e. . A right group action is a presheaf on the category created from a group , i.e. a category with one object and invertible morphisms.
Properties When is a small category, the functor category is cartesian closed. The poset 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.)
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 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 .
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.
See also Topos Category of elements Simplicial presheaf (this notion is obtained by replacing "set" with "simplicial set") Presheaf with transfers
Notes
References
Further reading
Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.