nLab ModCat



Categories of categories

Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\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 Cat_{Adj} :

ModCatCat AdjCat. ModCat \longrightarrow Cat_{Adj} \longrightarrow Cat \,.

Therefore a pseudofunctor Cat\mathcal{B} \longrightarrow Cat which factors through ModCatModCat 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)(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:

Last revised on September 29, 2023 at 16:59:17. See the history of this page for a list of all contributions to it.