nLab
functors and comma categories

This entry is about special properties of functors on comma categories.

Presheaves on over-categories and over-categories of presheaves

Let $C$ be a category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$. Write $\mathrm{PSh}(C/c)=[(C/C{)}^{\mathrm{op}},\mathrm{Set}]$ for the category of presheaves on $C/c$ and write $\mathrm{PSh}(C)/Y(y)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y:C\to \mathrm{PSh}(c)$ is the Yoneda embedding.

Proposition

There is an equivalence of categories

$$e:\mathrm{PSh}(C/c)\stackrel{\simeq }{\to }\mathrm{PSh}(C)/Y(c).$$e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.

Proof: The functor $e$ takes $F\in \mathrm{PSh}(C/c)$ to the presheaf $F\prime :d\mapsto {\bigsqcup }_{f\in C(d,c)}F(f)$ which is equipped with the natural transformation $\eta :F\prime \to Y(c)$ with component map ${\eta }_d{\bigsqcup }_{f\in C(d,c)}F(f)\to C(d,c)$.

A weak inverse of $e$ is given by the functor

$$\overline{e}:\mathrm{PSh}(C)/Y(c)\to \mathrm{PSh}(C/c)$$\bar e : PSh(C)/Y(c) \to PSh(C/c)

which sends $\eta :F\prime \to Y(C))$ to $F\in \mathrm{PSh}(C/c)$ given by

$$F:(f:d\to c)\mapsto F\prime (d){\mid }_c,$$F : (f : d \to c) \mapsto F'(d)|_c \,,

where $F\prime (d){\mid }_c$ is the pullback

$$\begin{array}{ccc}F\prime (d){\mid }_c& \to & F\prime (d)\\ \downarrow & & {\downarrow }^{{\eta }_d}\\ \mathrm{pt}& \stackrel{f}{\to }& C(d,c)\end{array}.$$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.

Example

Suppose the presheaf $F\in \mathrm{PSh}(C/c)$ does not actually depend on the morphsims to $C$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:

$$F:(C/c{)}^{\mathrm{op}}\to C^{\mathrm{op}}\to \mathrm{Set}.$$F : (C/c)^{op} \to C^{op} \to Set \,.

Then $F\prime (d)={\bigsqcup }_{f\in C(d,c)}F(f)={\bigsqcup }_{f\in C(d,c)}F(d)\simeq C(d,c)\times F(d)$ and hence $F\prime =Y(c)\times F$ with respect to the closed monoidal structure on presheaves.