Presheaf (category theory)

Functor from a category's opposite category to Set

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.[1]

## Examples

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* 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 elements 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 elements 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 element and invertible morphisms.

## Properties

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* When  C {\displaystyle C} is a small category, the functor category  C ^ = S e t C o p {\displaystyle {\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}} is cartesian closed.
* The poset of subobjects of  P {\displaystyle P} form a Heyting algebra, whenever  P {\displaystyle P} is an object of  C ^ = S e t C o p {\displaystyle {\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}} for small  C {\displaystyle C} .
* For any morphism  f : X → Y {\displaystyle f:X\to Y} of  C ^ {\displaystyle {\widehat {C}}} , the pullback functor of subobjects  f ∗ : S u b C ^ ( Y ) → S u b C ^ ( X ) {\displaystyle f^{*}:\mathrm {Sub} _{\widehat {C}}(Y)\to \mathrm {Sub} _{\widehat {C}}(X)} has a right adjoint, denoted  ∀ f {\displaystyle \forall _{f}} , and a left adjoint,  ∃ f {\displaystyle \exists _{f}} . These are the universal and existential quantifiers.
* A locally small category  C {\displaystyle C} embeds fully and faithfully into the category  C ^ {\displaystyle {\widehat {C}}} of set-valued presheaves via the Yoneda embedding which to every object  A {\displaystyle A} of  C {\displaystyle C} associates the hom functor C ( − , A ) {\displaystyle C(-,A)} .
* The category  C ^ {\displaystyle {\widehat {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 ^ {\displaystyle {\widehat {C}}} is the colimit completion of  C {\displaystyle C} (see #Universal property below.)

## Universal property

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The construction  C ↦ C ^ = F c t ( C op , S e t ) {\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. Then each functor  η : C → D {\displaystyle \eta :C\to D} factorizes as

C ⟶ y C ^ ⟶ η ~ D {\displaystyle C{\overset {y}{\longrightarrow }}{\widehat {C}}{\overset {\widetilde {\eta }}{\longrightarrow }}D}

where y is the Yoneda embedding and  η ~ : C ^ → D {\displaystyle {\widetilde {\eta }}:{\widehat {C}}\to D} is a, unique up to isomorphism, colimit-preserving functor called the Yoneda extension of  η {\displaystyle \eta } .

Proof: Given a presheaf F, by the density theorem, we can write  F = lim → ⁡ y U i {\displaystyle F=\varinjlim yU_{i}} where  U i {\displaystyle U_{i}} are objects in C. Then let  η ~ F = lim → ⁡ η U i , {\displaystyle {\widetilde {\eta }}F=\varinjlim \eta U_{i},} which exists by assumption. Since  lim → − {\displaystyle \varinjlim -} is functorial, this determines the functor  η ~ : C ^ → D {\displaystyle {\widetilde {\eta }}:{\widehat {C}}\to D} . Succinctly,  η ~ {\displaystyle {\widetilde {\eta }}} is the left Kan extension of  η {\displaystyle \eta } along y; hence, the name "Yoneda extension". To see  η ~ {\displaystyle {\widetilde {\eta }}} commutes with small colimits, we show  η ~ {\displaystyle {\widetilde {\eta }}} is a left-adjoint (to some functor). Define  H o m ( η , − ) : D → C ^ {\displaystyle {\mathcal {H}}om(\eta ,-):D\to {\widehat {C}}} to be the functor given by: for each object M in D and each object U in C,

H o m ( η , M ) ( U ) = Hom D ⁡ ( η U , M ) . {\displaystyle {\mathcal {H}}om(\eta ,M)(U)=\operatorname {Hom} _{D}(\eta U,M).}

Then, for each object M in D, since  H o m ( η , M ) ( U i ) = Hom ⁡ ( y U i , H o m ( η , M ) ) {\displaystyle {\mathcal {H}}om(\eta ,M)(U_{i})=\operatorname {Hom} (yU_{i},{\mathcal {H}}om(\eta ,M))} by the Yoneda lemma, we have:

Hom D ⁡ ( η ~ F , M ) = Hom D ⁡ ( lim → ⁡ η U i , M ) = lim ← ⁡ Hom D ⁡ ( η U i , M ) = lim ← ⁡ H o m ( η , M ) ( U i ) = Hom C ^ ⁡ ( F , H o m ( η , M ) ) , {\displaystyle {\begin{aligned}\operatorname {Hom} _{D}({\widetilde {\eta }}F,M)&=\operatorname {Hom} _{D}(\varinjlim \eta U_{i},M)=\varprojlim \operatorname {Hom} _{D}(\eta U_{i},M)=\varprojlim {\mathcal {H}}om(\eta ,M)(U_{i})\\\&=\operatorname {Hom} _{\widehat {C}}(F,{\mathcal {H}}om(\eta ,M)),\end{aligned}}}

which is to say  η ~ {\displaystyle {\widetilde {\eta }}} is a left-adjoint to  H o m ( η , − ) {\displaystyle {\mathcal {H}}om(\eta ,-)} .  ◻ {\displaystyle \square }

The proposition yields several corollaries. For example, the proposition implies that the construction  C ↦ C ^ {\displaystyle C\mapsto {\widehat {C}}} is functorial: i.e., each functor  C → D {\displaystyle C\to D} determines the functor  C ^ → D ^ {\displaystyle {\widehat {C}}\to {\widehat {D}}} .

## Variants

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Main article: presheaf on an ∞-category

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 → C ^ {\displaystyle C\to {\widehat {C}}} 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.[6]

## See also

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

## Notes

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1. ^ co-Yoneda lemma at the nLab
2. ^ Kashiwara & Schapira 2005, Corollary 2.4.3.
3. ^ Kashiwara & Schapira 2005, Proposition 2.7.1.
4. ^ Lurie, Definition 1.2.16.1.
5. ^ Lurie, Proposition 5.1.3.1.
6. ^ "copresheaf". nLab. Retrieved 4 September 2024.

## References

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* Kashiwara, Masaki; Schapira, Pierre (2005). Categories and sheaves. Grundlehren der mathematischen Wissenschaften. Vol. 332\. Springer. ISBN 978-3-540-27950-1.
* Lurie, J. Higher Topos Theory.
* Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Springer. ISBN 0-387-97710-4.
* Awodey, Steve (2006). Category Theory. doi:10.1093/acprof:oso/9780198568612.001.0001. ISBN 978-0-19-856861-2.

## Further reading

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* Presheaf at the nLab
* category of presheaves at the nLab
* Free cocompletion at the nLab
* Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.