A calculus for flow categories

TL;DR

This paper introduces a calculus of moves for framed flow categories that preserves their stable homotopy type, enabling classification and manipulation of these categories in a systematic, often manual, way.

Contribution

It develops an algorithmic calculus for modifying framed flow categories while maintaining their stable homotopy type, with a focus on categories of homological width at most three.

Findings

Flow categories with the same stable homotopy type are move equivalent if their homological width is at most three.

The calculus can be performed manually, facilitating practical applications.

Provides a method for classifying flow categories based on their stable homotopy types.

Abstract

We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The process we describe is essentially algorithmic and can often be performed by hand, without the aid of a computer program.