A homotopy theory of weak {\omega}-categories

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.