nLab
functors and comma categories

This entry is about special properties of functors on comma categories. See also category of presheaves.

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

$${\eta }_d:{\bigsqcup }_{f\in C(d,c)}F(f)\to C(d,c):((f\in C(d,c),\theta \in F(f))\mapsto f.$$\eta_d : \sqcup_{f \in C(d,c)} F(f) \to C(d,c) : ((f \in C(d,c), \theta \in F(f)) \mapsto f \,.

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.

See over-topos for more.

Over-categories of presheaf categories and presheaves on categories of elements

Generalizing the above,

Proposition

For every $P:{\mathrm{Set}}^D$, there is an equivalence of categories

$$\phi :{\mathrm{Set}}^{\mathrm{el}(P)}\stackrel{\simeq }{\to }{\mathrm{Set}}^D/P.$$\varphi : \mathbf{Set}^{el(P)} \stackrel{\simeq}{\to} \mathbf{Set}^D / P \,.

where $\mathrm{el}(P)=*/P$ is the category of elements of $P$.

Proof

The construction is completely analogous to the above; Given $F:{\mathrm{Set}}^{\mathrm{el}(P)}$, $\phi (F)$ is defined pointwise as a coproduct:

$$\phi (F)(c)=\coprod _{(x,\mathrm{Pc})}F(x,\mathrm{Pc})$$\varphi(F)(c) = \coprod_{(x,Pc)} F(x,Pc)

where $(x,\mathrm{Pc})=(*\stackrel{x}{\to }\mathrm{Pc})$ is an object of $\mathrm{el}(P)$. The action on morphisms is defined analogously. This comes equipped with a natural transformation $\alpha :\phi (F)\to P$, with component

$$\begin{array}{c}{\alpha }_c:\phi (F)(c)\to \mathrm{Pc}\\ {\alpha }_c(F(x,\mathrm{Pc}))=x\in \mathrm{Pc}\\ \end{array}$$\array{ \alpha_c \colon \varphi(F)(c) \to Pc \\ \alpha_c(F(x,Pc)) = x \in Pc \\ }

Given an object $(Q\stackrel{\alpha }{\to }P)$ of ${\mathrm{Set}}^D/P$ the action of a weak inverse $\overline{\phi }$ can be specified as $\overline{\phi }(\alpha )(x,\mathrm{Pc})={\alpha }_c^{-1}(x)$, that is, the wedge of the pullback:

$$\begin{array}{ccc}\overline{\phi }(\alpha )(x,\mathrm{Pc})& \to & \mathrm{Qc}\\ \downarrow & & {\downarrow }^{{\alpha }_c}\\ \mathrm{pt}& \stackrel{x}{\to }& \mathrm{Pc}\end{array}.$$\array{ \bar{\varphi}(\alpha)(x,Pc) &\to& Qc \\ \downarrow && \downarrow^{\alpha_c} \\ pt &\stackrel{x}{\to}& Pc } \,.

The action of $\overline{\phi }(\alpha )$ on arrows of $\mathrm{el}(P)$, functoriality, etc is derived from its definition as a pullback and the def of morphisms in $\mathrm{el}(P)$.

Relationship with the over-categories statement

Putting $D=C^{\mathrm{op}},P=Y(c)$ in the above yields:

$${\mathrm{Set}}^{\mathrm{el}(Y(c))}\simeq {\mathrm{Set}}^D/Y(c)$$\mathbf{Set}^{el(Y(c))} \simeq \mathbf{Set}^D / Y(c)

Now it is easy to see that $\mathrm{el}(Y(c))\simeq (C/c{)}^{\mathrm{op}}$; we get then:

$${\mathrm{Set}}^{(C/c{)}^{\mathrm{op}}}\simeq {\mathrm{Set}}^{C^{\mathrm{op}}}/Y(c)$$\mathbf{Set}^{(C / c)^{op}} \simeq \mathbf{Set}^{C^{op}} / Y(c)

In higher category theory

For the analogous result in the context of (∞,1)-category theory see (∞,1)-Category of (∞,1)-presheaves -- Interaction with overcategories