In category theory, a branch of mathematics, a presheaf on a category
{\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {Set} }
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 {\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}}
and it is called the category of presheaves on
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 C\mapsto {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}
is called the colimit completion of C because of the following universal property: Proposition[3] — Let C, D be categories and assume D admits small colimits.
factorizes as where y is the Yoneda embedding and
is a, unique up to isomorphism, colimit-preserving functor called the Yoneda extension of
Proof: Given a presheaf F, by the density theorem, we can write
which exists by assumption.
is functorial, this determines the functor
is the left Kan extension of
along y; hence, the name "Yoneda extension".
commutes with small colimits, we show
is a left-adjoint (to some functor).
by the Yoneda lemma, we have: which is to say
The proposition yields several corollaries.
For example, the proposition implies that the construction
A presheaf of spaces on an ∞-category C is a contravariant functor from C to the ∞-category of spaces (for example, the nerve of the category of CW-complexes.
)[4] It is an ∞-category version of a presheaf of sets, as a "set" is replaced by a "space".
The notion is used, among other things, in the ∞-category formulation of Yoneda's lemma that says:
is fully faithful (here C can be just a simplicial set.
)[5] A copresheaf of a category C is a presheaf of Cop.
In other words, it is a covariant functor from C to Set.