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
There are several versions of a (very large) 2-category of model categories, depending on which notion of transformation of adjoints one takes to be the 2-morphisms between 1-morphisms given by Quillen functors.
One choice is to consider 2-morphisms to be conjugate transformations of adjoints between Quillen adjunctions [Hovey (1999), p. 24, cf. also Harpaz & Prasma (2015), Def. 2.5.3], such that forgetting the model category-structure is a forgetful 2-functor to $Cat_{Adj}$:
Therefore a pseudofunctor $\mathcal{B} \longrightarrow Cat$ which factors through $ModCat$ this way has as Grothendieck construction a bifibration of model categories. Under good conditions, the domain of this bifibration carries itself an induced model category structure, see at model structure on Grothendieck constructions.
$(n+1,r+1)$-categories of (n,r)-categories
The 2-category of model categories, left-pointing Quillen adjunctions and conjugate transformations of adjoints is considered in:
Mark Hovey, p. 24 of: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]
Yonatan Harpaz, Matan Prasma, Def. 2.5.3 in: The Grothendieck construction for model categories, Advances in Mathematics 281 (2015) 1306-1363 [arXiv:1404.1852, 10.1016/j.aim.2015.03.031]
Last revised on September 29, 2023 at 16:59:17. See the history of this page for a list of all contributions to it.