An integral model structure and truncation theory for coherent group actions

TL;DR

This paper develops a comprehensive homotopy-theoretic framework for coherent group actions, introducing a model category that unifies all Segal group actions and establishing a truncation theory with applications to Postnikov towers.

Contribution

It constructs a global model category for Segal group actions and proves a rectification and truncation theory within this framework, advancing the understanding of homotopy actions.

Findings

Constructed a model category for all Segal group actions.

Proved a global rectification result for these actions.

Developed a truncation theory with a characterization of n-truncated actions.

Abstract

In this work we study the homotopy theory of coherent group actions from a global point of view, where we allow both the group and the space acted upon to vary. Using the model of Segal group actions and the model categorical Grothendieck construction we construct a model category encompassing all Segal group actions simultaneously. We then prove a global rectification result in this setting. We proceed to develop a general truncation theory for the model-categorical Grothendieck construction and apply it to the case of Segal group actions. We give a simple characterization of $n$-truncated Segal group actions and show that every Segal group action admits a convergent Postnikov tower.