What is "presheaf"

n. (context category theory English) A contravariant functor whose domain is a category whose objects are open sets of a topological space and whose morphisms are inclusion mappings.David Jao. "sheaf" (version 11). PlanetMath.org. Freely available at http://planetmath.org/?op=getobj;from=objects;id=2878

In category theory, a branch of mathematics, a presheaf on a category C is a functor F: C → Set. If C 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 Ĉ = Set. A functor into Ĉ 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 F: C → V as a V-valued presheaf.