nLab
presheaf

Definition

A presheaf on a category $C$ is nothing but a functor

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

where $C^{\mathrm{op}}$ is the opposite category of $C$. More generally, given any category $S$, an $S$-valued presheaf on $C$ is a functor

$$C^{\mathrm{op}}\to S$$C^{op} \to S

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

• There are certain special contexts in which one calls functors $F:C^{\mathrm{op}}\to S$ ‘presheaves’ instead of just functors, namely:

  • 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 of presheaves

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.

Every presheaf is a colimit of representable presheaves.

An elegant way to express this for any preaheaf $F:C^{\mathrm{op}}\to \mathrm{Set}$ is as 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)

using Yoneda reduction.

Another way to express the same is as follows: let $Y:C\to [C^{\mathrm{op}},\mathrm{Set}]$ be the Yoneda embedding and let $C_F$ be the corresponding comma category

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

and let $p:C_F\to C$ the canonical functor. Then

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

To see this notice that for every $B\in [C^{\mathrm{op}},\mathrm{Set}]$ and using the property of the Hom 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. But the last term is seen by inspection to be equivalent to

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

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