Category Theory of Symbolic Dynamics

TL;DR

This paper applies category theory to symbolic dynamics, analyzing subshifts and block maps to identify universal objects, classify morphisms, and explore categorical properties like completeness and regularity.

Contribution

It introduces a categorical framework for symbolic dynamics, solves the dual Extension Lemma, and investigates conserved quantities via coequalizers.

Findings

Identification of natural categories with subshifts and block maps

Solutions to the dual Extension Lemma

Results on conserved quantities and categorical properties

Abstract

We study the central objects of symbolic dynamics, that is, subshifts and block maps, from the perspective of basic category theory, and present several natural categories with subshifts as objects and block maps as morphisms. Our main goals are to find universal objects in these symbolic categories, to classify their block maps based on their category theoretic properties, and to establish as many natural properties (finite completeness, regularity etc.) as possible. Existing definitions in category theory suggest interesting new problems for block maps. Our main technical contributions are the solution to the dual problem of the Extension Lemma and results on certain types of conserved quantities, suggested by the concept of a coequalizer.