Presheaf (category theory)

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In category theory, 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. Maps between presheaves are called profunctors.

The category $\mathbf{Set}^{C^{op}}$ of presheaves over $C$ is often written $\hat{C}$ .

[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 $C$ .