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In category theory, a branch of mathematics, a $V$ -valued presheaf $F$ on a category $C$ is a functor $F : C^{o p} \to 𝐕$ . 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, and is an example of a functor category. It is often written as $\hat{C} = {𝐒 𝐞 𝐭}^{C^{o p}}$ . A functor into $\hat{C}$ is sometimes called a profunctor.

A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some object A of C is called a representable presheaf.

Examples

When $C$ is a small category, the functor category $\hat{C} = {𝐒 𝐞 𝐭}^{C^{o p}}$ is cartesian closed.
The partially ordered set of subobjects of $P$ form a Heyting algebra, whenever $P$ is an object of $\hat{C} = {𝐒 𝐞 𝐭}^{C^{o p}}$ for small $C$ .
For any morphism $f : X \to Y$ of $\hat{C}$ , the pullback functor of subobjects $f^{*} : {S u b}_{\hat{C}} (Y) \to {S u b}_{\hat{C}} (X)$ has a right adjoint, denoted $\forall_{f}$ , and a left adjoint, $\exists_{f}$ . These are the universal and existential quantifiers.
A locally small category $C$ embeds fully and faithfully into the category $\hat{C}$ of set-valued presheaves 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 $C$ .

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Saunders Mac Lane, Ieke Moerdijk, "Sheaves in Geometry and Logic" (1992) Springer-Verlag ISBN 0-387-97710-4