Presheaf (category theory)

Short description: Functor from a category's opposite category to Set

In category theory, a branch of mathematics, a presheaf on a category C is a functor F:Cop->𝐒𝐞𝐭. 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 on C into a category, and is an example of a functor category. It is often written as C^=𝐒𝐞𝐭Cop. A functor into 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:Cop->𝐕 as a 𝐕-valued presheaf.[1]

## Contents

* 1 Examples
* 2 Properties
* 3 Universal property
* 4 Variants
* 5 See also
* 6 Notes
* 7 References
* 8 Further reading

## Examples

* A simplicial set is a Set-valued presheaf on the simplex category C=Δ.

## Properties

* When C is a small category, the functor category C^=𝐒𝐞𝐭Cop is cartesian closed.
* The poset of subobjects of P form a Heyting algebra, whenever P is an object of C^=𝐒𝐞𝐭Cop for small C.
* For any morphism f:X->Y of C^, the pullback functor of subobjects f*:SubC^(Y)->SubC^(X) has a right adjoint, denoted ∀f, and a left adjoint, ∃f. These are the universal and existential quantifiers.
* A locally small category C embeds fully and faithfully into the category C^ of set-valued presheaves via the Yoneda embedding which to every object A of C associates the hom functor C(−,A).
* The category C^ admits small limits and small colimits.[2] See limit and colimit of presheaves for further discussion.
* The density theorem states that every presheaf is a colimit of representable presheaves; in fact, C^ is the colimit completion of C (see #Universal property below.)

## Universal property

The construction C↦C^=𝐅𝐜𝐭(Cop,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. Then each functor η:C->D factorizes as

C⟶yC^⟶η~D

where y is the Yoneda embedding and η~:C^->D is a, unique up to isomorphism, colimit-preserving functor called the Yoneda extension of η.

Proof: Given a presheaf F, by the density theorem, we can write F=lim->yUi where Ui are objects in C. Then let η~F=lim->ηUi, which exists by assumption. Since lim->− is functorial, this determines the functor η~:C^->D. Succinctly, η~ is the left Kan extension of η along y; hence, the name "Yoneda extension". To see η~ commutes with small colimits, we show η~ is a left-adjoint (to some functor). Define ℋom(η,−):D->C^ to be the functor given by: for each object M in D and each object U in C,

ℋom(η,M)(U)=HomD(ηU,M).

Then, for each object M in D, since ℋom(η,M)(Ui)=Hom⁡(yUi,ℋom(η,M)) by the Yoneda lemma, we have:

HomD(η~F,M)=HomD(lim->ηUi,M)=lim<-HomD(ηUi,M)=lim<-ℋom(η,M)(Ui)=HomC^(F,ℋom(η,M)),

which is to say η~ is a left-adjoint to ℋom(η,−). ◻

The proposition yields several corollaries. For example, the proposition implies that the construction C↦C^ is functorial: i.e., each functor C->D determines the functor C^->D^.

## Variants

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: C->PShv(C) is fully faithful (here C can be just a simplicial set.)[5]

## See also

* Topos
* Category of elements
* Simplicial presheaf (this notion is obtained by replacing "set" with "simplicial set")
* Presheaf with transfers

## Notes

1. ↑ co-Yoneda lemma in nLab
2. ↑ Kashiwara & Schapira 2005, Corollary 2.4.3. harvnb error: no target: CITEREFKashiwaraSchapira2005 (help)
3. ↑ Kashiwara & Schapira 2005, Proposition 2.7.1. harvnb error: no target: CITEREFKashiwaraSchapira2005 (help)
4. ↑ Lurie, Definition 1.2.16.1. harvnb error: multiple targets (2×): CITEREFLurie (help)
5. ↑ Lurie, Proposition 5.1.3.1. harvnb error: multiple targets (2×): CITEREFLurie (help)

## References

* {{Cite book

| 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

* Lurie, J.. Higher Topos Theory.
* Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Springer. ISBN 0-387-97710-4.

## Further reading

* Presheaf in nLab
* Free cocompletion in nLab
* Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.

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