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

Mateusz Przybylski, Marian Mrozek, Jim Wiseman

TL;DR

This paper provides an algorithmic framework for classifying isomorphism classes in the Szymczak category Szym(FRel) for finite sets with binary relations, a foundational step toward Conley theory for relations in discrete-time dynamics. It proves that every Endo(FRel) object is Szym-isomorphic to a canonical form and that canonical objects are classified by their Endo(FRel) type, reducing the problem to graph-theoretic data. The authors introduce tools such as gdom, gim, Inv, and classifying graphs to capture inter-component connections and periods, and demonstrate nontriviality and limitations of these invariants through examples. They also connect the FRel case to shift equivalence of Boolean matrices and show how the FSet case specializes via periodic-point reductions, with implications for multivalued dynamics and potential expansions to linear-relations settings and Conley-type indices.

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.

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

TL;DR

This paper provides an algorithmic framework for classifying isomorphism classes in the Szymczak category Szym(FRel) for finite sets with binary relations, a foundational step toward Conley theory for relations in discrete-time dynamics. It proves that every Endo(FRel) object is Szym-isomorphic to a canonical form and that canonical objects are classified by their Endo(FRel) type, reducing the problem to graph-theoretic data. The authors introduce tools such as gdom, gim, Inv, and classifying graphs to capture inter-component connections and periods, and demonstrate nontriviality and limitations of these invariants through examples. They also connect the FRel case to shift equivalence of Boolean matrices and show how the FSet case specializes via periodic-point reductions, with implications for multivalued dynamics and potential expansions to linear-relations settings and Conley-type indices.

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.
Paper Structure (12 sections, 57 theorems, 120 equations, 7 figures, 1 table)

This paper contains 12 sections, 57 theorems, 120 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Szymczak functor
  5. Categories.
  6. Category of endomorphisms.
  7. Szymczak category.
  8. Szymczak functor in $\text{\sc FSet}$
  9. Szymczak functor in $\text{\sc FRel}$
  10. Objects in canonical form
  11. Classifying graphs
  12. Final remarks

Key Result

Theorem 2.1

Every object in $\operatorname{Endo}(\text{\sc FRel})$ is isomorphic in $\text{\sc Szym}(\text{\sc FRel})$ to a canonical object.

Figures (7)

  • Figure 1: Canonical objects in $\text{\sc Szym}(\text{\sc FRel})$ of cardinality zero (empty relation), one and two.
  • Figure 2: Canonical objects in $\text{\sc Szym}(\text{\sc FRel})$ of cardinality three with three strongly connected components.
  • Figure 3: Canonical objects in $\text{\sc Szym}(\text{\sc FRel})$ of cardinality three with less than three strongly connected components.
  • Figure 4: Two isomorphic relations in Szymczak's category. The eventual period $p$ and the period $q$ of the relation on the left are both $p=q=3$. The equivalence classes of the relation $\sim_R$ are marked with colors.
  • Figure 5: Relations $R_1$, $R_2$ and $R_3$ (from left to right) isomorphic in Szymczak's category. Only relation $R_3$ is in canonical form.
  • ...and 2 more figures

Theorems & Definitions (57)

  • Theorem 2.1: see Theorem \ref{['thm:each-obj-has-its-canonical-form']}
  • Theorem 2.2: see Theorem \ref{['thm:szym-isom-induces-can-obj-endo-isom']}
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Proposition 4.4
  • Proposition 4.5
  • ...and 47 more