Presheaf (category theory)

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