Homotopy theory of stricter $n$-categories

Abstract

We make strict $n$-categories even stricter by requiring they satisfy higher exchange laws governed by Hadzihasanovic's theory of regular directed complexes. We study the first properties of stricter $n$-categories, in particular, we define the Gray product, and prove stability under suspension, which is non-trivial. After reviewing and briefly expanding the theory diagrammatic sets and their associated model structures for $(\infty, n)$-categories, we construct a folk model structure on stricter $n$-categories, show that the walking equivalence coincides with the stricter polygraph generated by the walking equivalence in diagrammatic sets, and finally, that the folk model structure on stricter $n$-categories is right transferred from the diagrammatic model structure along a nerve construction.