The Szymczak Functor on the Category of Finite Sets and Finite Relations

TL;DR

This paper introduces an algorithmic classification of the Szymczak functor's isomorphism classes over finite sets with relations, advancing the foundation for Conley theory in relation-based dynamical systems.

Contribution

It provides the first algorithmizable classification of the Szymczak functor's isomorphism classes in the context of finite sets and relations, facilitating Conley theory development.

Findings

Classification of isomorphism classes achieved

Algorithmic approach developed for the Szymczak functor

Foundation laid for Conley theory for relations

Abstract

The Szymczak functor is a tool used to construct the Conley index for dynamical systems with discrete time. We present an algorithmizable classification of isomorphism classes in the Szymczak category over the category of finite sets with arbitrary relations as morphisms. The research is the first step towards the construction of Conley theory for relations.