Presheaf (category theory)
In category theory, a branch of mathematics, a -valued presheaf on a category is a functor . Often presheaf is defined to be a Set-valued presheaf. 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 into a category, and is an example of a functor category. It is often written as . A functor into is sometimes called a profunctor.
- 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-set .
- The presheaf category is (up to equivalence of categories) the free colimit completion of the category .