category theory

# Contents

## 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 $\mathrm{PSh}\left(C/c\right)=\left[\left(C/c{\right)}^{\mathrm{op}},\mathrm{Set}\right]$ for the category of presheaves on $C/c$ and write $\mathrm{PSh}\left(C\right)/Y\left(y\right)$ for the over category of presheaves on $C$ over the presheaf $Y\left(c\right)$, where $Y:C\to \mathrm{PSh}\left(c\right)$ is the Yoneda embedding.

###### Proposition

There is an equivalence of categories

$e:\mathrm{PSh}\left(C/c\right)\stackrel{\simeq }{\to }\mathrm{PSh}\left(C\right)/Y\left(c\right)\phantom{\rule{thinmathspace}{0ex}}.$e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.
###### Proof

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

${\eta }_{d}:{\bigsqcup }_{f\in C\left(d,c\right)}F\left(f\right)\to C\left(d,c\right):\left(\left(f\in C\left(d,c\right),\theta \in F\left(f\right)\right)↦f\phantom{\rule{thinmathspace}{0ex}}.$\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

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

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

$F:\left(f:d\to c\right)↦F\prime \left(d\right){\mid }_{c}\phantom{\rule{thinmathspace}{0ex}},$F : (f : d \to c) \mapsto F'(d)|_c \,,

where $F\prime \left(d\right){\mid }_{c}$ is the pullback

$\begin{array}{ccc}F\prime \left(d\right){\mid }_{c}& \to & F\prime \left(d\right)\\ ↓& & {↓}^{{\eta }_{d}}\\ \mathrm{pt}& \stackrel{f}{\to }& C\left(d,c\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.
###### Example

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

$F:\left(C/c{\right)}^{\mathrm{op}}\to {C}^{\mathrm{op}}\to \mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$F : (C/c)^{op} \to C^{op} \to Set \,.

Then $F\prime \left(d\right)={\bigsqcup }_{f\in C\left(d,c\right)}F\left(f\right)={\bigsqcup }_{f\in C\left(d,c\right)}F\left(d\right)\simeq C\left(d,c\right)×F\left(d\right)$ and hence $F\prime =Y\left(c\right)×F$ 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

$\phi :{\mathrm{Set}}^{\mathrm{el}\left(P\right)}\stackrel{\simeq }{\to }{\mathrm{Set}}^{D}/P\phantom{\rule{thinmathspace}{0ex}}.$\varphi : \mathbf{Set}^{el(P)} \stackrel{\simeq}{\to} \mathbf{Set}^D / P \,.

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

###### Proof

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

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

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

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

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

$\begin{array}{ccc}\overline{\phi }\left(\alpha \right)\left(x,\mathrm{Pc}\right)& \to & \mathrm{Qc}\\ ↓& & {↓}^{{\alpha }_{c}}\\ \mathrm{pt}& \stackrel{x}{\to }& \mathrm{Pc}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \bar{\varphi}(\alpha)(x,Pc) &\to& Qc \\ \downarrow && \downarrow^{\alpha_c} \\ pt &\stackrel{x}{\to}& Pc } \,.

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

### Relationship with the over-categories statement

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

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

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

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

Revised on December 6, 2011 05:37:03 by Toby Bartels (64.89.53.227)