An integral model structure and truncation theory for coherent group 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.

The Grothendieck construction for model categories

“…For such a theory to be widely applicable one would like to be able to work in a setting of coherent group actions. This will be undertaken thoroughly in a subsequent paper [11] using Segal group actions as developed in [21]. In this subsection we will content with presenting a strict and a weak model for group actions and using Theorem 4.1.3 to relate the two.…”