Presheaf (category theory)

Presheaf (category theory)

In category theory, a branch of mathematics, a V-valued presheaf F on a category C is a functor F:C^mathrm{op} omathbf{V}. Often "presheaf" is defined to be a Set-valued presheaf. 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, often written hat{C}. A functor into hat{C} is sometimes called a profunctor.


* A category C embeds fully and faithfully into the category hat{C} of set-valued presheaves via the Yoneda embedding mathrm{Y}_c which to every object A of C associates the hom-set C(-,A).
* The presheaf category hat{C} is (up to equivalence of categories) the free colimit completion of the category C.

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