Presheaf (category theory)

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

Examples

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

Properties

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

Universal property

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

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

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

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

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

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

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

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

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: $C\to PShv(C)$ is fully faithful (here C can be just a simplicial set.)^[5]

Notes

↑ co-Yoneda lemma in nLab
↑ Kashiwara–Schapira, Corollary 2.4.3.
↑ Kashiwara–Schapira, Proposition 2.7.1.
↑ Lurie, Definition 1.2.16.1.
↑ Lurie, Proposition 5.1.3.1.

References

Kashiwara, Masaki; Schapira, Pierre (2006). Categories and sheaves.
Lurie, J. Higher Topos Theory
Saunders Mac Lane, Ieke Moerdijk, "Sheaves in Geometry and Logic" (1992) Springer-Verlag ISBN 0-387-97710-4