In category theory, a branch of mathematics, a presheaf on a category C {\displaystyle C} is a functor F : C o p → S e t {\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {Set} } . If C {\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 C {\displaystyle C} into a category, and is an example of a functor category. It is often written as C ^ = S e t C o p {\displaystyle {\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}} and it is called the category of presheaves on C {\displaystyle C} . A functor into C ^ {\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 F : C o p → V {\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {V} } as a V {\displaystyle \mathbf {V} } -valued presheaf.
Examples A simplicial set is a Set-valued presheaf on the simplex category C = Δ {\displaystyle C=\Delta } . A directed multigraph is a presheaf on the category with two objects and two parallel morphisms between them i.e. C = ( E ⟶ t s V ) {\displaystyle C=(E{\overset {s}{\underset {t}{\longrightarrow }}}V)} . An arrow category is a presheaf on the category with two objects and one morphism between them. i.e. C = ( E ⟶ f V ) {\displaystyle C=(E{\overset {f}{\longrightarrow }}V)} . A right group action is a presheaf on the category created from a group G {\displaystyle G} , i.e. a category with one object and invertible morphisms.
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