Model structures for diagrammatic $(\infty, n)$-categories

TL;DR

This paper develops new model structures for diagrammatic $( , n)$-categories, establishing their properties and equivalences, and proves the homotopy hypothesis for a specific $ty$-category model.

Contribution

It constructs novel model structures for diagrammatic $( , n)$-categories and proves their equivalences, including the homotopy hypothesis for $ty$-groupoids.

Findings

Constructed model structures for diagrammatic $( , n)$-categories.

Proved Quillen equivalences between marked and unmarked models.

Established the homotopy hypothesis for a model of $ty$-groupoids.

Abstract

Diagrammatic sets admit a notion of internal equivalence in the sense of coinductive weak invertibility, with similar properties to its analogue in strict $\omega$-categories. We construct a model structure whose fibrant objects are diagrammatic sets in which every round pasting diagram is equivalent to a single cell -- its weak composite -- and propose them as a model of $(\infty, \infty)$-categories. For each $n < \infty$, we then construct a model structure whose fibrant objects are those $(\infty, \infty)$-categories whose cells in dimension $> n$ are all weakly invertible. We show that weak equivalences between fibrant objects are precisely morphisms that are essentially surjective on cells of all dimensions. On the way to this result, we also construct model structures for $(\infty, n)$-categories on marked diagrammatic sets, which split into a coinductive and an inductive case when $n = \infty$, and prove that they are Quillen equivalent to the unmarked model structures when $n < \infty$ and in the coinductive case of $n = \infty$. Finally, we prove that the $(\infty, 0)$-model structure is Quillen equivalent to the classical model structure on simplicial sets. This establishes the first proof of the homotopy hypothesis for a model of $\infty$-groupoids defined as $(\infty, \infty)$-categories whose cells in dimension $> 0$ are all weakly invertible.