Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category ${\displaystyle C}$ is a functor ${\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {Set} }$. If ${\displaystyle C}$ 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 ${\displaystyle C}$ into a category, and is an example of a functor category. It is often written as ${\displaystyle {\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}}$. A functor into ${\displaystyle {\widehat {C}}}$ 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 ${\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {V} }$ as a ${\displaystyle \mathbf {V} }$-valued presheaf.[1]

Examples

• A simplicial set is a Set-valued presheaf on the simplex category ${\displaystyle C=\Delta }$.

Properties

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

Universal property

The construction ${\displaystyle C\mapsto {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}$ 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 ${\displaystyle \eta :C\to D}$ factorizes as

${\displaystyle C{\overset {y}{\longrightarrow }}{\widehat {C}}{\overset {\widetilde {\eta }}{\longrightarrow }}D}$

where y is the Yoneda embedding and ${\displaystyle {\widetilde {\eta }}:{\widehat {C}}\to D}$ is a colimit-preserving functor called the Yoneda extension of ${\displaystyle \eta }$.

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

${\displaystyle {\mathcal {H}}om(\eta ,M)(U)=\operatorname {Hom} _{D}(\eta U,M).}$

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

${\displaystyle {\begin{aligned}\operatorname {Hom} _{D}({\widetilde {\eta }}F,M)&=\operatorname {Hom} _{D}(\varinjlim \eta U_{i},M)=\varprojlim \operatorname {Hom} _{D}(\eta U_{i},M)=\varprojlim {\mathcal {H}}om(\eta ,M)(U_{i})\\&=\operatorname {Hom} _{\widehat {C}}(F,{\mathcal {H}}om(\eta ,M)),\end{aligned}}}$

which is to say ${\displaystyle {\widetilde {\eta }}}$ is a left-adjoint to ${\displaystyle {\mathcal {H}}om(\eta ,-)}$. ${\displaystyle \square }$

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

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: ${\displaystyle C\to PShv(C)}$ 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, Corollary 2.4.3. harvnb error: no target: CITEREFKashiwara–Schapira (help)
3. Kashiwara–Schapira, Proposition 2.7.1. harvnb error: no target: CITEREFKashiwara–Schapira (help)
4. Lurie, Definition 1.2.16.1. harvnb error: no target: CITEREFLurie (help)
5. Lurie, Proposition 5.1.3.1. harvnb error: no target: CITEREFLurie (help)

References

• Kashiwara, Masaki; Schapira, Pierre (2006). Categories and sheaves.CS1 maint: ref=harv (link)
• Lurie, J. Higher Topos Theory
• Saunders Mac Lane, Ieke Moerdijk, "Sheaves in Geometry and Logic" (1992) Springer-Verlag 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.