Presheaf (category theory)

Template:Short description In category theory, a branch of mathematics, a presheaf on a category $C$ is a functor $F:C^{\mathrm{o}\mathrm{p}}\to 𝐒𝐞𝐭$. 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 $\hat{C}={𝐒𝐞𝐭}^{C^{\mathrm{o}\mathrm{p}}}$ and it is called the category of presheaves on $C$. A functor into $\hat{C}$ 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 $F:C^{\mathrm{o}\mathrm{p}}\to 𝐕$ as a $𝐕$-valued presheaf.

Examples

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

Properties

• When $C$ is a small category, the functor category $\hat{C}={𝐒𝐞𝐭}^{C^{\mathrm{o}\mathrm{p}}}$ is cartesian closed.
• The poset of subobjects of $P$ form a Heyting algebra, whenever $P$ is an object of $\hat{C}={𝐒𝐞𝐭}^{C^{\mathrm{o}\mathrm{p}}}$ for small $C$.
• For any morphism $f:X\to Y$ of $\hat{C}$, the pullback functor of subobjects $f^{*}:{\mathrm{S}\mathrm{u}\mathrm{b}}_{\hat{C}}(Y)\to {\mathrm{S}\mathrm{u}\mathrm{b}}_{\hat{C}}(X)$ has a right adjoint, denoted ${\mathrm{\forall }}_f$, and a left adjoint, ${\mathrm{\exists }}_f$. These are the universal and existential quantifiers.
• A locally small category $C$ embeds fully and faithfully into the category $\hat{C}$ of set-valued presheaves via the Yoneda embedding which to every object $A$ of $C$ associates the hom functor $C(-,A)$.
• The category $\hat{C}$ admits small limits and small colimits. See limit and colimit of presheaves for further discussion.
• The density theorem states that every presheaf is a colimit of representable presheaves; in fact, $\hat{C}$ is the colimit completion of $C$ (see #Universal property below.)

Universal property

The construction $C\mapsto \hat{C}=𝐅𝐜𝐭(C^{\text{op}},𝐒𝐞𝐭)$ is called the colimit completion of C because of the following universal property:

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

$\mathscr{H}om(\eta ,M)(U)={\mathrm{Hom}}_D(\eta U,M).$

Then, for each object M in D, since $\mathscr{H}om(\eta ,M)(U_i)=\mathrm{Hom}(y{U}_i,\mathscr{H}om(\eta ,M))$ by the Yoneda lemma, we have:

$\begin{array}{cc}{\mathrm{Hom}}_D(\tilde{\eta }F,M)& ={\mathrm{Hom}}_D(\underrightarrow{\lim }\eta U_i,M)=\underleftarrow{\lim }{\mathrm{Hom}}_D(\eta U_i,M)=\underleftarrow{\lim }\mathscr{H}om(\eta ,M)(U_i)\\ & ={\mathrm{Hom}}_{\hat{C}}(F,\mathscr{H}om(\eta ,M)),\end{array}$

which is to say $\tilde{\eta }$ is a left-adjoint to $\mathscr{H}om(\eta ,-)$. $◻$

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

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.) 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.)

A copresheaf of a category C is a presheaf of C^op. In other words, it is a covariant functor from C to Set.

See also

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

References

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Further reading

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• Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.