Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category ${\displaystyle C}$ is a functor ${\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {Set} }$. If ${\displaystyle C}$ 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 on ${\displaystyle C}$ into a category, and is an example of a functor category. It is often written as ${\displaystyle {\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}}$. A functor into ${\displaystyle {\widehat {C}}}$ 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.

Some authors refer to a functor ${\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {V} }$ as a ${\displaystyle \mathbf {V} }$-valued presheaf.^[1]