model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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.
flavors of higher toposes
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.
The terminology “model topos” is due to:
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:
reviewed in
Last revised on February 15, 2023 at 15:43:24. See the history of this page for a list of all contributions to it.