In category theory, a $V$-valued presheaf $F$ on a category $C$ is a functor $F:C^\backslash mathrm\{op\}\backslash to\backslash mathbf\{V\}$. One often simply talk of a presheaf to mean a Set-valued presheaf. Maps between presheaves are called profunctors.

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

Properties

• A category $C$ embeds fully and faithfuly in $\backslash hat\{C\}$ via the Yoneda embedding $\backslash mathrm\{Y\}\_c$ which to every object $A$ of $C$ associates the hom-set $C(-,A)$.
• The presheaf category $\backslash hat\{C\}$ is (up to equivalence of categories) the free colimit completion of the category $C$.

See also

• presheaf (mathematics)

Category theory