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 $F:C^mathrm\left\{op\right\} omathbf\left\{V\right\}$. 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 $hat\left\{C\right\}$. A functor into $hat\left\{C\right\}$ is sometimes called a profunctor.

Properties

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

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