model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -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 is a model topos if there is a simplicial site and a Quillen equivalence to the local model structure on sSet-presheaves over .
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 -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.