Topologizing infinite quivers and their mutations

TL;DR

This paper introduces topological spaces of infinite quivers, studies their properties, and characterizes convergence of infinite mutation sequences, highlighting a special 'Fra"issé quiver' and relating to prior topological models.

Contribution

It defines new topological spaces for infinite quivers, analyzes their properties, and characterizes mutation sequence convergence, including a novel 'Fra"issé quiver' example.

Findings

Two spaces are homeomorphic to the Baire space for countably infinite vertex sets

Complete characterization of convergence and divergence domains of infinite mutation sequences in one space

Identification of a special 'Fra"issé quiver' illustrating differences between finite and infinite mutations

Abstract

We define several topological spaces whose points are quivers with a given infinite vertex set $X$. In the special case when $X$ is countably infinite, we show that two of the spaces of interest are homeomorphic to the Baire space $\mathbb{N}^\mathbb{N}$. We study properties of countably infinite quivers as subspaces of these topological spaces and prove a ``meta-theorem'' about hereditary properties of quivers. Furthermore, we approach the question of convergence for infinite mutation sequences in these spaces, providing a complete characterization of the (non-)density of the domains of convergence and divergence of infinite mutation sequences in one of these spaces and a partial characterization in the other. We then draw attention to a very special infinite quiver which we call the \emph{Fra\"iss\'e quiver} that draws a clear contrast between the behavior of finite and infinite mutation sequences. Finally, we reproduce (a very mild modification of) a previously-constructed topological space due to Ervin and Jackson as a subquotient of one of the spaces of interest.