An inductive model structure for strict $\infty$-categories

TL;DR

This paper develops a new semi-model category framework for strict ∞-categories, connecting it to existing models and exploring multiple notions of weak (∞,∞)-categories.

Contribution

It constructs a semi-model category for marked strict ∞-categories, relates it to known models via localization and nerve functors, and discusses multiple notions of (∞,∞)-categories.

Findings

A semi-model category for marked strict ∞-categories is constructed.

The model structure relates to the Verity model via a Quillen adjunction.

At least three distinct notions of (∞,∞)-categories are identified.

Abstract

We construct a left semi-model category of "marked strict $\infty$-categories" for which the fibrant objects are those whose marked arrows satisfy natural closure properties and are weakly invertible. The canonical model structure on strict $\infty$-categories can be recovered as a left Bousfield localization of this model structure. We show that an appropriate extension of the Street nerve to the marked setting produces a Quillen adjunction between our model category and the Verity model structure for complicial sets, generalizing previous results by the second named author. Finally, we use this model structure to study, in the setting of strict $\infty$-categories, the idea that there are several non-equivalent notions of weak $(\infty,\infty)$-categories - depending on what tower of $(\infty,n)$-categories is used. We show that there ought to be at least three different notions of $(\infty,\infty)$-categories.