# nLab model topos

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

A model topos is a model category that presents an (∞,1)-topos.

###### Definition

A model category $\mathcal{C}$ is a model topos if there is a simplicial site $K$ and a Quillen equivalence $\mathcal{C} \simeq sPSh(K)_{loc}$ to the local model structure on sSet-presheaves over $K$.

This appears as Rezk, 6.1.

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

$\hookrightarrow$ | accessible categories | | model category theory | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left 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 |

## References

Created on October 15, 2012 17:05:45 by Urs Schreiber (82.113.99.246)