nLab
model topos

Idea

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

Definition

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

This appears as Rezk, 6.1.

Related concepts

• model site, model structure on simplicial presheaves

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

(n,r)-categories…	satisfying Giraud's axioms	inclusion of left exaxt localizations	generated under colimits from small objects		localization of free cocompletion		generated under filtered colimits from small objects
(0,1)-category theory	(0,1)-toposes	$\hookrightarrow$	algebraic lattices	$\simeq$ Porst’s theorem	subobject lattices in accessible reflective subcategories of presheaf categories
category theory	toposes	$\hookrightarrow$	locally presentable categories	$\simeq$ Adámek-Rosický’s theorem	accessible reflective subcategories of presheaf categories	$\hookrightarrow$	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

• Charles Rezk, Toposes and homotopy toposes (pdf)