An inductive model structure for strict ∞-categories

Abstract

We construct a left semi-model category of “marked strict ∞-categories” for which the fibrant objects are those whose marked arrows satisfy natural closure properties and are invertible up to higher marked arrows. The canonical model structure on strict ∞-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 ∞-categories, the idea that, because they are two different “truncation functors” taking an $(\infty ,n)$ to an $(\infty ,n-1)$-category, there are two non-equivalent definitions for the $(\infty ,1)$-category of $(\infty ,\infty )$-categories as a limit of the $(\infty ,1)$-categories of $(\infty ,n)$-categories. We show that in fact there seem to be at least three non-equivalent ways of constructing an $(\infty ,1)$-category of $(\infty ,\infty )$-categories.