nLab
model topos

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.

Related concepts

• model site, model structure on simplicial presheaves

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Adámek-Rosický‘s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

• Charles Rezk, Toposes and homotopy toposes, 2010 (pdf)