nLab
presheaf

Definition

A presheaf on a category $C$ is a functor

$$F:C^{\mathrm{op}}\to \mathrm{Set}$$F : C^{op} \to Set

from the opposite category $C^{\mathrm{op}}$ of $C$ to the category Set of sets. Equivalently this may be thought of as a contravariant functor $F:C\to \mathrm{Set}$.

More generally, given any category $S$, an $S$-valued presheaf on $C$ is a functor

$$F:C^{\mathrm{op}}\to S.$$F : C^{op} \to S.

Historically, the initial applications of presheaves and sheaves involved cases like $S=\mathrm{CRing}$ (the category of commutative rings), $S=$Ab, $S=R$-$\mathrm{Mod}$, etc. Later, especially with the development of topos theory, the primary importance of the category of set-valued (pre)sheaves as topos was recognized; these other cases could be considered algebraic objects which live in the topos. This article and the one on sheaf topos recognize these later developments by making the set-valued case the default (in other words, presheaf or sheaf without further qualification is understood to refer to the set-valued case).

The category of presheaves on $C$, usually denoted ${\mathrm{Set}}^{C^{\mathrm{op}}}$ or $[C^{\mathrm{op}},\mathrm{Set}]$, but often abbreviated as $\hat{C}$, has:

• functors $F:C^{\mathrm{op}}\to \mathrm{Set}$ as objects;

• natural transformations between such functors as morphisms.

As such, it is an example of a functor category.

Remarks

• Speaking of functors as presheaves indicates operations that one wants to do apply to these functors, or certain properties that one wants to check.

  • when $S=\mathrm{Set}$, and especially one is interested in the Yoneda embedding of a category $C$ into its presheaf category $[C^{\mathrm{op}},\mathrm{Set}]$ for purposes of studying, for instance, limits, colimits, ind-objects, and pro-objects of $C$;

  • or when there is the structure of a site on $C$, such that it makes sense to ask if a given presheaf is actually a sheaf.

• One generally useful way to think of presheaves is in the sense of space and quantity.

• In the case where $S=\mathrm{Set}$ and $C$ is small, an important general principle is that the presheaf category $[C^{\mathrm{op}},\mathrm{Set}]$ is the free cocompletion of $C$. Intuitively, it is formed by taking $C$ and ‘freely throwing in small colimits’. The category $C$ is contained in $[C^{\mathrm{op}},\mathrm{Set}]$ via the Yoneda embedding

  $$Y:C\to [C^{\mathrm{op}},\mathrm{Set}]$$Y : C \to [C^{op},Set]

  The Yoneda embedding sends each object $c\in C$ to the presheaf

  $$F(-)=\mathrm{hom}(-,c)$$F(-) = hom(-, c)

  Presheaves of this form, or isomorphic to those of this form, are called representable; among their properties, representable presheaves always turn colimits into limits, in the sense that a representable functor from $C^{\mathrm{op}}$ to $\mathrm{Set}$ turns colimits in $C$ (i.e., limits in $C^{\mathrm{op}}$) into limits in $\mathrm{Set}$ (i.e., colimits in ${\mathrm{Set}}^{\mathrm{op}}$). In general, such continuity is a necessary but not sufficient criterion for representability; however, nicely enough, it is sufficient when $C$ itself is a presheaf category. To see this, suppose $K$ is such a presheaf on $C=[D^{\mathrm{op}},\mathrm{Set}]$, and let $G=KY$, a presheaf on $D$. By the Yoneda lemma, we have a natural isomorphism between $[D^{\mathrm{op}},\mathrm{Set}](Y(-),G)$ and $KY(-)$. But by the free cocompletion property of the Yoneda embedding, a colimit-preserving functor on presheaves is entirely determined by its precomposition with $Y$; accordingly, our isomorphism must extend to an identification of $[C^{\mathrm{op}},\mathrm{Set}](-,G)$ with $K(-)$, thus establishing the representability of $K$.

Properties

Limits and colimits

Any category of presheaves is complete and cocomplete, with both limits and colimits being computed pointwise. That is, to compute the limit or colimit of a diagram $F:D\to {\mathrm{Set}}^{C^{\mathrm{op}}}$, we think of it as a functor $F:D\times C^{\mathrm{op}}\to \mathrm{Set}$ and take the limit or colimit in the $D$ variable.

Proposition

Every presheaf is a colimit of representable presheaves.

An elegant way to express this colimit for a presheaf $F:C^{\mathrm{op}}\to \mathrm{Set}$ is in terms of the coend identity

$$F(-)={\int }^{c\in C}F(c)\times {\mathrm{hom}}_C(-,c),$$F(-) = \int^{c \in C} F(c) \times hom_C(-,c) \,,

which follows by Yoneda reduction. See also co-Yoneda lemma.

More concretely: let $Y:C\to [C^{\mathrm{op}},\mathrm{Set}]$ denote the Yoneda embedding and let $C_F:=Y/F$ be the corresponding comma category, the category of elements of $F$:

$$C_F:=\left\{\begin{array}{ccccc}Y(V)& & \stackrel{Y(g)}{\to }& & Y(V\prime )\\ & {}_f\searrow & & {\swarrow }_{f\prime }\\ & & F\end{array}\right\}$$C_F := \left\lbrace \array{ Y(V) &&\stackrel{Y(g)}{\to}&& Y(V') \\ & {}_f\searrow && \swarrow_{f'} \\ && F } \right\rbrace

and let $p:C_F\to C$ the canonical forgetful functor. Then the colimit over representables expression $F$ is

$$F\simeq {\mathrm{colim}}_{(Y(V)\to F)\in C_F}(Y\circ p).$$F \simeq colim_{(Y(V) \to F) \in C_F} (Y\circ p) \,.

This is often written with some convenient abuse of notation as

$$F\simeq {\mathrm{colim}}_{V\to F}V.$$F \simeq colim_{V \to F} V \,.

Notice that these formulas can also be understood as those for the left Kan extension (see there) of $F$ along the identity functor.

Proof

Notice that for every $B\in [C^{\mathrm{op}},\mathrm{Set}]$ and using the property of the hom-functor we have

$$\begin{array}{rl}{\mathrm{Hom}}_{[C^{\mathrm{op}},\mathrm{Set}]}({\mathrm{colim}}_{(Y(V)\to F)\in C_F}(Y\circ p),B)& \simeq {\lim }_{(Y(V)\to F)\in C_F}{\mathrm{Hom}}_{[C^{\mathrm{op}},\mathrm{Set}]}(Y(V),B)\\ & \simeq {\lim }_{(Y(V)\to F)\in C_F}B(V)\end{array}$$\begin{aligned} Hom_{[C^{op}, Set]}(colim_{(Y(V) \to F) \in C_F} (Y\circ p),B) &\simeq lim_{(Y(V) \to F) \in C_F} Hom_{[C^{op}, Set]}(Y(V),B) \\ & \simeq lim_{(Y(V) \to F) \in C_F} B(V) \end{aligned}

by the Yoneda lemma.

By the definition of limit we have that

$$\cdots ={\mathrm{Hom}}_{[C_F^{\mathrm{op}},\mathrm{Set}]}(\mathrm{pt},B),$$\cdots=Hom_{[C_F^{op}, Set]}(pt,B),

so for each natural transformation $\alpha \in {\mathrm{Hom}}_{[C_F^{\mathrm{op}},\mathrm{Set}]}(\mathrm{pt},B)$ and each object $h:Y(V)\to F\in C_F$, ${\alpha }_h$ is a map $\{*\}\to B(V)$, that is, it is an element of $B(V)$. However, by Yoneda, we know that each object $h:Y(V)\to F\in C_F$ specifies a unique element $h\in F(V)$. Then rephrasing this, $\alpha$ specifies a function $F(V)\to B(V)$. The naturality of this assignment is guaranteed by the naturality of the map $\alpha$. Then $\alpha$ induces a natural transformation $k^{\alpha }:F\to B$. It’s easy to check that $k$ defines an isomorphism:

$${\mathrm{Hom}}_{[C_F^{\mathrm{op}},\mathrm{Set}]}(\mathrm{pt},B)\simeq {\mathrm{Hom}}_{[C^{\mathrm{op}},\mathrm{Set}]}(F,B).$$Hom_{[C_F^{op}, Set]}(pt,B) \simeq Hom_{[C^{op}, Set]}(F,B) \,.

Since this holds for all $B$, the claim follows, again using the Yoneda lemma.

Special cases

• representable presheaf

• concrete presheaf

Examples

Examples for presheaves are abundant. Here is a non-representative selection of some examples.

• For $C$ a locally small category, every object $c\in C$ gives rise to the representable presheaf ${\mathrm{Hom}}_C(-,c):C^{\mathrm{op}}\to \mathrm{Set}$.

• More generally, for $i:C\hookrightarrow D$ a subcategory of a locally small category $D$, every object $d\in D$ gives rise to the presheaf

  $${\mathrm{Hom}}_D(i(-),D):C^{\mathrm{op}}\to \mathrm{Set}.$$Hom_D(i(-), D) : C^{op} \to Set \,.

  Let’s spell this out in more detail: given a mophism $\varphi :V\to U$ in $C$, we can take any morphism $f:i(U)\to X$ in ${\mathrm{Hom}}_D(U,X)$ and turn it into a morphism $V\stackrel{f}{\to }U\stackrel{\varphi }{\to }X$ in ${\mathrm{Hom}}_D(i(V),X)$. This determines a map of set

  $$f^{*}:{\mathrm{Hom}}_D(i(U),X)\to {\mathrm{Hom}}_D(i(V),X).$$f^* : Hom_{D}(i(U),X) \to Hom_{D}(i(V),X) \,.

  So we have a functorial assignment of the form

  $$\begin{array}{ccccc}W& & \mapsto & & {\mathrm{Hom}}_{\mathrm{Diff}}(i(W),X)\\ {\downarrow }^g& & & & {\uparrow }^{g^{*}}\\ V& & \mapsto & & {\mathrm{Hom}}_{\mathrm{Diff}}(i(V),X)\\ {\downarrow }^f& & & & {\uparrow }^{f^{*}}\\ U& & \mapsto & & {\mathrm{Hom}}_{\mathrm{Diff}}(i(U),X)\end{array}.$$\array{ W && \mapsto && Hom_{Diff}(i(W),X) \\ \downarrow^g &&&& \uparrow^{g^*} \\ V && \mapsto && Hom_{Diff}(i(V),X) \\ \downarrow^f &&&& \uparrow^{f^*} \\ U && \mapsto && Hom_{Diff}(i(U),X) } \,.

  Of course $i$ here could be any functor whatsoever. Asking if such a presheaf is representable is asking for a right adjoint functor of $i$.

• A simplicial set is a presheaf on the simplex category

  A globular set is a presheaf on the globe category.

  A cubical set is a presheaf on the cube category.

• A diffeological space is a concrete presheaf on CartSp.

• An important class of presheaves is those on a category of open subsets $\mathrm{Op}(X)$ of a topological space or smooth manifold $X$.

  Traditional standard examples include: the presheaf of smooth functions on $X$, that assigns to each $U\subset X$ the set $C^{\mathrm{\infty }}(C,ℝ)$ of smooth functions and to each unclusion $V\subset U$ the corresponding restriction operation of functions.

… etc. pp.

Related concepts

• presheaf

  constant presheaf

• (2,1)-presheaf

• (∞,1)-presheaf

• (∞,n)-presheaf