Presheaf (category theory)
- 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, often written . A functor into is sometimes called a profunctor.
Properties
* A 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 .
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