In category theory, a branch of mathematics, a presheaf on a category [math]\displaystyle{ C }[/math] is a functor [math]\displaystyle{ F\colon C^\mathrm{op}\to\mathbf{Set} }[/math]. If [math]\displaystyle{ C }[/math] 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 [math]\displaystyle{ C }[/math] into a category, and is an example of a functor category. It is often written as [math]\displaystyle{ \widehat{C} = \mathbf{Set}^{C^\mathrm{op}} }[/math]. A functor into [math]\displaystyle{ \widehat{C} }[/math] 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 [math]\displaystyle{ F\colon C^\mathrm{op}\to\mathbf{V} }[/math] as a [math]\displaystyle{ \mathbf{V} }[/math]-valued presheaf.[1]
The construction [math]\displaystyle{ C \mapsto \widehat{C} = \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}) }[/math] 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. Then each functor [math]\displaystyle{ \eta: C \to D }[/math] factorizes as
where y is the Yoneda embedding and [math]\displaystyle{ \widetilde{\eta}: \widehat{C} \to D }[/math] is a, unique up to isomorphism, colimit-preserving functor called the Yoneda extension of [math]\displaystyle{ \eta }[/math].
Proof: Given a presheaf F, by the density theorem, we can write [math]\displaystyle{ F =\varinjlim y U_i }[/math] where [math]\displaystyle{ U_i }[/math] are objects in C. Then let [math]\displaystyle{ \widetilde{\eta} F = \varinjlim \eta U_i, }[/math] which exists by assumption. Since [math]\displaystyle{ \varinjlim - }[/math] is functorial, this determines the functor [math]\displaystyle{ \widetilde{\eta}: \widehat{C} \to D }[/math]. Succinctly, [math]\displaystyle{ \widetilde{\eta} }[/math] is the left Kan extension of [math]\displaystyle{ \eta }[/math] along y; hence, the name "Yoneda extension". To see [math]\displaystyle{ \widetilde{\eta} }[/math] commutes with small colimits, we show [math]\displaystyle{ \widetilde{\eta} }[/math] is a left-adjoint (to some functor). Define [math]\displaystyle{ \mathcal{H}om(\eta, -): D \to \widehat{C} }[/math] to be the functor given by: for each object M in D and each object U in C,
Then, for each object M in D, since [math]\displaystyle{ \mathcal{H}om(\eta, M)(U_i) = \operatorname{Hom}(y U_i, \mathcal{H}om(\eta, M)) }[/math] by the Yoneda lemma, we have:
which is to say [math]\displaystyle{ \widetilde{\eta} }[/math] is a left-adjoint to [math]\displaystyle{ \mathcal{H}om(\eta, -) }[/math]. [math]\displaystyle{ \square }[/math]
The proposition yields several corollaries. For example, the proposition implies that the construction [math]\displaystyle{ C \mapsto \widehat{C} }[/math] is functorial: i.e., each functor [math]\displaystyle{ C \to D }[/math] determines the functor [math]\displaystyle{ \widehat{C} \to \widehat{D} }[/math].
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: [math]\displaystyle{ C \to PShv(C) }[/math] is fully faithful (here C can be just a simplicial set.)[5]
| last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | title=Categories and sheaves | year=2005 |url=https://books.google.com/books?id=mc5DAAAAQBAJ |publisher=Springer |isbn=978-3-540-27950-1 |volume=332 |series=Grundlehren der mathematischen Wissenschaften
Original source: https://en.wikipedia.org/wiki/Presheaf (category theory).
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