Presheaf (category theory)

Functor from a category's opposite category to Set
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.

A simplicial set is a Set valued presheaf on the simplex category .

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.)

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.)