• Images
• Videos
• Documents
• Books
• Articles

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 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

• 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

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:

Proposition³—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

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 → P S h v ( C ) {\displaystyle C\to PShv(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

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

Notes

• 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

• 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.

References

1. co-Yoneda lemma at the nLab https://ncatlab.org/nlab/show/co-Yoneda+lemma ↩

2. Kashiwara & Schapira 2005, Corollary 2.4.3. - Kashiwara, Masaki; Schapira, Pierre (2005). Categories and sheaves. Grundlehren der mathematischen Wissenschaften. Vol. 332. Springer. ISBN 978-3-540-27950-1. https://books.google.com/books?id=mc5DAAAAQBAJ ↩

3. Kashiwara & Schapira 2005, Proposition 2.7.1. - Kashiwara, Masaki; Schapira, Pierre (2005). Categories and sheaves. Grundlehren der mathematischen Wissenschaften. Vol. 332. Springer. ISBN 978-3-540-27950-1. https://books.google.com/books?id=mc5DAAAAQBAJ ↩

4. Lurie, Definition 1.2.16.1. - Lurie, J. Higher Topos Theory. ↩

5. Lurie, Proposition 5.1.3.1. - Lurie, J. Higher Topos Theory. ↩

6. "copresheaf". nLab. Retrieved 4 September 2024. https://ncatlab.org/nlab/show/copresheaf ↩