Irreversible homotopy and a notion of irreversible Lusternik-Schnirelmann category

TL;DR

This paper introduces a new framework for modeling irreversible processes using ir-homotopy and Lusternik-Schnirelmann ir-category, establishing their properties and invariance under ir-homotopy in $T_0$ spaces.

Contribution

It develops a novel notion of irreversibility in topology, defining ir-paths, ir-homotopy, and ir-category, and proves invariance of ir-category under ir-homotopy equivalence.

Findings

Irreversible paths and homotopies are defined in $T_0$ spaces.

Lusternik-Schnirelmann ir-category is an invariant under ir-homotopy.

The framework models irreversible phenomena in topological terms.

Abstract

This work was intended as an attempt to investigate a model of irreversible process and natural phenomena. For this, we introduce the notion of irreversible path (that for brevity we write ir-path), ir-homotopy, ir-contractible space, and Lusternik-Schnirelmann ir-category by equipping the $I=[0,1]$ with left order topology. We will restrict the irreversibility of definitions to $T_0$ Spaces, such that for $T_1$ spaces, the ir-paths are constant. After providing some theorems and properties of these notions, eventually, we prove that Lusternik-Schnirelmann ir-category is an invariant of ir-homotopy equivalence.