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

• Ho(CombModCat)

flavors of higher toposes

• 1-topos

  • (0,1)-topos

  • (1,1)-topos = topos

  • (2,1)-topos

  • $(n,1)$-topos

  • (∞,1)-topos ($n$-localic)

    $\;\;\;\;$ model topos

• 2-topos

  • (2,2)-topos ($n$-localic)

  • (∞,2)-topos

• $n$-topos

  • $(\infty,n)$-topos

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

The terminology “model topos” is due to:

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

Essentially this idea appears earlier in:

• Carlos Simpson, A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi (arXiv:math/9903167)

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry I: Topos theory, Advances in Mathematics 193 2 (2005) 257-372 [arXiv:math.AG/0207028, doi:10.1016/j.aim.2004.05.004]

In the context of categorical semantics for univalent homotopy type theory, the combination of terminology “model topos” with type-theoretic model category to type-theoretic model topos:

• Michael Shulman, §1.3 in: All $(\infty,1)$-toposes have strict univalent universes (arXiv:1904.07004)

reviewed in

• Emily Riehl, §6.1 of: On the $\infty$-topos semantics of homotopy type theory, lecture at Logic and higher structures CIRM (Feb. 2022) [pdf, pdf]