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Concept
Presheaf (category theory)
Summary
In , a branch of mathematics, a presheaf on a 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 . 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 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 .
When is a , 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 and small colimits. See limit and colimit of presheaves for further discussion.
The 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:
Proof: Given a presheaf F, by the , 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 .
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