# nLab category of presheaves

### Context

#### Topos Theory

Could not include topos theory - contents

category theory

# Contents

## Definition

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

$PSh(C) := [C^{op}, Set]$

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 $PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$ and write $PSh(C)/Y(y)$ 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 $\eta_d \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.

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.

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

### Models in presheaf toposes

See at models in presheaf toposes.

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

## References

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$\hookrightarrow$algebraic lattices$\simeq$ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes$\hookrightarrow$locally presentable categories$\simeq$ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories$\hookrightarrow$accessible categories
model category theorymodel toposes$\hookrightarrow$combinatorial model categories$\simeq$ Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes$\hookrightarrow$locally presentable (∞,1)-categories$\simeq$
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories$\hookrightarrow$accessible (∞,1)-categories

Revised on April 17, 2014 21:05:06 by Adeel Khan (132.252.249.179)