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 $PSh(C/c) = [(C/c)^{op}, Set]$ 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.

Proposition

There is an equivalence of categories

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

Proof

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

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

Example

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.

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

Generalizing the above,

Proposition

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 $P$.

Proof

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 $el(P)$.

Relationship with the over-categories statement

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:

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

In higher category theory

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

Related concepts

• comma category

• presheaf