Symbolic dynamics and the category of graphs

TL;DR

This paper develops a homotopy theory framework for directed graphs based on symbolic dynamics, introducing invariants like the basal graph that capture walk structures and are preserved under homotopy.

Contribution

It establishes a Quillen model structure on graphs, defines a homotopy category, and introduces the basal graph as a new homotopy invariant that refines existing graph invariants.

Findings

The basal graph is a homotopy invariant.

Homotopy equivalences preserve walk space topologies.

Basal graphs are finer invariants than zeta series.

Abstract

Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, $C^*$-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which the weak equivalences are those graph morphisms which induce bijections on the set of walks. We determine the resulting homotopy category. We also introduce a "finite-level" homotopy category which respects the natural topology on the set of walks. To each graph we associate a basal graph, well defined up to isomorphism. We show that the basal graph is a homotopy invariant for our model structure, and that it is a finer invariant than the zeta series of a finite graph. We also show that, for finite walkable graphs, if $B$ is basal and separated then the walk spaces for $X$ and $B$ are topologically conjugate if and only if $X$ and $B$ are homotopically equivalent for our model structure.