functors and comma categories

This entry is about special properties of functors on comma categories. See also category 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 $PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$ and write $PSh(C)/Y(c)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the Yoneda embedding.

There is an equivalence of categories

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

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map

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

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

$\bar e : PSh(C)/Y(c) \to PSh(C/c)$

which sends $\eta : F' \to Y(c)$ to $F \in PSh(C/c)$ given by

$F : (f : d \to c) \mapsto F'(d)|_c
\,,$

where $F'(d)|_c$ is the pullback

$\array{
F'(d)|_c &\to& F'(d)
\\
\downarrow && \downarrow^{\eta_d}
\\
pt &\stackrel{f}{\to}& C(d,c)
}
\,.$

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

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

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

See over-topos for more.

Generalizing the above,

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

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

where $el(P) = \ast / P$ is the category of elements of $P$.

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

$\varphi(F)(c) = \coprod_{(x,Pc)} F(x,Pc)$

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

$\array{
\alpha_c \colon \varphi(F)(c) \to Pc \\
\alpha_c(F(x,Pc)) = x \in Pc \\
}$

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

$\array{
\bar{\varphi}(\alpha)(x,Pc) &\to& Qc
\\
\downarrow && \downarrow^{\alpha_c}
\\
pt &\stackrel{x}{\to}& Pc
}
\,.$

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

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

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

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

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

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

Last revised on July 30, 2021 at 00:24:57. See the history of this page for a list of all contributions to it.