nLab ModCat

Idea

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 $Cat_{Adj}$:

$$ModCat \longrightarrow Cat_{Adj} \longrightarrow Cat \,.$$

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.

Related concepts

• double category of model categories

• Ho(CombModCat)

• Grothendieck construction for model categories

$(n+1,r+1)$-categories of (n,r)-categories

• Pos

• Set

  • Rel

• Grpd, ∞Grpd

• Cat

  • Ho(Cat)

  • AccCat

  • PrCat

  • LexCat

  • MonCat

  • VCat

  • CatAdj

  • Prof

  • Operad

• 2Cat

• ModCat, CombModCat

• (∞,1)Cat

  • Pr(∞,1)Cat

  • (∞,1)Operad

• (∞,n)Cat

References

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]