on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A model topos is a model category that presents an (∞,1)-topos.
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.
(n,r)-categories… | satisfying Giraud's axioms | inclusion of left exact 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 |