A homotopy theory of weak {\omega}-categories

TL;DR

This paper investigates a model structure for weak -categories using cellular sets, analyzing localizers, stability issues, and fibrant objects, and provides examples challenging the conjectured model's validity.

Contribution

It introduces a new approach to modeling weak -categories via cellular sets, examines stability of localizers, and presents counterexamples to the conjectured model structure.

Findings

Minimal localizer containing spine inclusions is not stable under two-point suspension.

Explicit example of a nontrivial contractible cofibrant strict -category.

The fibrant objects for the proposed model structure are not trivially fibrant.

Abstract

In this paper, we consider the model structure on the category of cellular sets originally conjectured by Cisinski and Joyal to give a model for the homotopy theory of weak (\omega)-categories. We demonstrate first that any (\Theta)-localizer containing the spine inclusions (\iota: \Sp[t] \hookrightarrow \Theta[t]) must also contain the maps (X\times \iota: X\times \Sp[t] \hookrightarrow X\times \Theta[t]) for all objects ([t]) of (\Theta) and all cellular sets (X). This implies in particular that a cellular set (S) is local with respect to the set of spine inclusions if and only if it is Cartesian-local. However, we show that the minimal localizer containing the spine inclusions is not stable under two-point suspension, which implies that the equivalences between objects fibrant for this model structure only depend on their height-(0) and height-(1) structure. We then try to see if adopting an approach similar to Rezk's, namely looking at all of the suspensions of the inclusion of a point into a freestanding isomorphism. We call the fibrant objects for this model structure \dfn{isostable Joyal-fibrant} cellular sets. We understand the resulting model structure to be conjectured by a few mathematicians to give a model structure for a category of weak (\omega)-categories. However, we make short work of this claim by producing an explicit example of a nontrivial contractible cofibrant strict (\omega)-category (with respect to the folk model structure) and showing that it is, first, not trivially fibrant, and second, proving that it is fibrant with respect to the isomorphism-stable Joyal model structure. We then speculate on a few of the possible ways to construct a localizer that does actually have our desired properties, leaving this question open for a future revision.