An Inductive Strategy Towards a Solution to the Generalized Homotopy Hypothesis

TL;DR

This paper introduces an inductive approach using distributive monads to construct an $( abla,0)$-coherator, aiming to prove the Generalized Homotopy Hypothesis through model structure transfers.

Contribution

It develops an inductive framework and necessary conditions for transferring model structures to support the proof of the Generalized Homotopy Hypothesis.

Findings

Constructed the inductive coherator for $ abla$-groupoids.

Provided conditions for successful transfer of model structures.

Showed that successively transferring model structures implies the hypothesis.

Abstract

Using the theory of distributive series of monads, we construct an $(\infty,0)$-coherator called the \emph{inductive coherator}. The category of models out of the inductive coherator serve as a model for $\infty$-groupoids that possess an underlying globular set. Once we establish the construction for the inductive coherator, we provide the framework for an inductive strategy to prove the Generalized Homotopy Hypothesis obtained by transferring model structure off of the category of $n$-groupoids onto the category of $(n+1)$-groupoids. Moreover, we provide a necessary and sufficient condition for the transfer of model structure to be successful. We conclude by showing if the transfer of model structure may be completed successively, then the Generalized Homotopy Hypothesis is true.