Presheaf (category theory)

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In category theory, a branch of mathematics, a $V$ -valued presheaf $F$ on a category $C$ is a functor $F:C^\mathrm{op}\to\mathbf{V}$ . One often simply talk of a presheaf to mean 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.

[edit] Properties

A category $C$ embeds fully and faithfuly in $\hat{C}$ via the Yoneda embedding $Y c$ 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 $C$ .