The category of cellular resolutions

TL;DR

This paper introduces the category of cellular resolutions, exploring its properties and establishing connections with homotopy colimits and Morse theory to better understand the structure of cellular resolutions.

Contribution

It defines the category of cellular resolutions and studies its properties, providing a framework for analyzing higher-level structures beyond specific families.

Findings

Lifts homotopy colimits from topology to the category

Shows Morse theory on cellular resolutions is compatible with the category

Provides foundational understanding of the structure of cellular resolutions

Abstract

Cellular resolutions is a well studied topic on the level of single resolutions and certain specific families of cellular resolutions. One question coming out of the work on families is to understand the structure of cellular resolutions more generally. We give a starting point to understanding higher level structures by defining the category of cellular resolutions. In this paper we study the properties of this category. The main results are in lifting homotopy colimits from topology and Morse theory on cellular resolution being compatible with the category.