# 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.

** Properties **

* 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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