Induced model structures for higher categories

TL;DR

This paper introduces a new criterion for establishing model structures via functors with adjoints, enabling the construction of models for higher categories such as $$-categories and $$-groupoids in various settings.

Contribution

It provides a novel criterion for left-induced model structures along functors with adjoints, broadening the toolkit for modeling higher categories.

Findings

Constructed new model structures on cubical sets, prederivators, marked simplicial sets, and simplicial spaces.

Demonstrated the criterion's applicability to models of $$-categories and $$-groupoids.

Extended the understanding of how adjunctions induce model structures in higher category theory.

Abstract

We give a new criterion guaranteeing existence of model structures left-induced along a functor admitting both adjoints. This works under the hypothesis that the functor induces idempotent adjunctions at the homotopy category level. As an application, we construct new model structures on cubical sets, prederivators, marked simplicial sets and simplicial spaces modeling $\infty$-categories and $\infty$-groupoids.