Interaction homotopy and interaction homology

TL;DR

This paper introduces an algebraic topology framework to analyze interactions between spaces, defining interaction spaces and their homology, with potential applications in complex systems analysis.

Contribution

It develops the concept of interaction spaces and demonstrates that their singular homology is invariant under interaction homotopy, advancing mathematical tools for studying complex interactions.

Findings

Interaction singular homology is an invariant under interaction homotopy.

The framework provides new algebraic topology tools for analyzing interactions.

Potential for practical applications in complex systems analysis.

Abstract

Interactions in complex systems are widely observed across various fields, drawing increased attention from researchers. In mathematics, efforts are made to develop various theories and methods for studying the interactions between spaces. In this work, we present an algebraic topology framework to explore interactions between spaces. We introduce the concept of interaction spaces and investigate their homotopy, singular homology, and simplicial homology. Furthermore, we demonstrate that interaction singular homology serves as an invariant under interaction homotopy. We believe that the proposed framework holds potential for practical applications.