Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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
The notion of model $(\infty,1)$-category (or model $\infty$-category, for short) is the $(\infty,1)$-categorification of that of model category.
Where the classical model structure on simplicial sets is an archetypical example of a model category, so simplicial $\infty$-groupoids (“simplicial spaces”, bisimplicial sets) form an archetypical example of a model $\infty$-category. In this example, a fundamental application of the theory says, for instance, that geometric realization preserves homotopy pullbacks of homotopy Kan fibrations (see there).
Aaron Mazel-Gee, Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces [arXiv:1412.8411]
Aaron Mazel-Gee, Model ∞-categories II: Quillen adjunctions, New York Journal of Mathematics 27 (2021) 508-550. [arXiv:1510.04392, nyjm:27-21]
Aaron Mazel-Gee, Model ∞-categories III: the fundamental theorem, New York Journal of Mathematics 27 (2021) 551-599 [arXiv:1510.04777, nyjm:27-22]
Last revised on October 15, 2023 at 08:39:51. See the history of this page for a list of all contributions to it.