nLab model (∞,1)-category

Idea

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).

References

• 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]