The topology of critical processes, III (Computing homotopy)

TL;DR

This paper extends directed algebraic topology to include critical processes, focusing on computing the fundamental category of controlled spaces with an eye toward understanding their homotopy structure.

Contribution

It introduces methods to compute the fundamental category of controlled spaces that model critical processes, advancing the understanding of non-reversible process spaces.

Findings

Methods for computing the fundamental category of controlled spaces.

Framework for modeling critical, indecomposable, and unstoppable processes.

Foundation for analyzing the homotopy structure in future work.

Abstract

Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The previous parts of this series introduced controlled spaces and their fundamental category. Here we study how to compute the latter. The homotopy structure of these spaces will be examined in Part IV.