topos theory

category theory

# Contents

## Definition

For $C$ a small category, its category of presheaves is the functor category

$\mathrm{PSh}\left(C\right):=\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$PSh(C) := [C^{op}, Set]

from the opposite category of $C$ to Set.

An object in this category is a presheaf. See there for more details.

## Properties

### Characterization

###### Theorem

A category $E$ is equivalent to a presheaf topos if and only if it is cocomplete, atomic, and regular.

This is due to Marta Bunge.

### Cartesian closed monoidal structure

As every topos, a category of presheaves is a cartesian closed monoidal category.

For details on the closed structure see

### 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 ${\eta }_{d}{\bigsqcup }_{f\in C\left(d,c\right)}F\left(f\right)\to C\left(d,c\right)$.

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 $\eta :F\prime \to Y\left(C\right)\right)$ 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.

For the analog statement in (∞,1)-category theory see

For (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves.

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes$↪$algebraic lattices$\simeq$ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes$↪$locally presentable categories$\simeq$ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories$↪$accessible categories
model category theorymodel toposes$↪$combinatorial model categories$\simeq$ Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes$↪$locally presentable (∞,1)-categories$\simeq$
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories$↪$accessible (∞,1)-categories

Revised on October 15, 2012 17:59:16 by Urs Schreiber (82.113.99.246)