The purpose of this thesis is to advance the study and application of the field of persistent homology through both categorical and quiver theoretic viewpoints. While persistent homology has its roots in these topics, there is a wealth of material that can still be offered up by using these familiar lenses at new angles.
There are four sections of results.
The first of these discusses a categorical framework for persistent homology that circumvents quiver theoretic structure, both in practice and in theory, by means of viewing the process as factored through a quotient category. In this chapter, the widely used persistent homology algorithm collectively known as reduction is laid out purely in terms of matrix multiplication and Bruhat reduction.
The remaining results rest on a quiver theoretic approach.
The second section focuses on an algebraic stability theorem for generalized persistence modules for a certain class of finite posets. Both the class of posets and their discretized nature are what make the results unique, while the structure is taken with heavy inspiration from the work of Ulrich Bauer and Michael Lesnick.
The third section deals with taking directed limits of posets and the subsequent expansion of categories to show that the discretized work in the second section recovers classical results over the continuum.
The fourth section presents a full algebraic stability theorem for arbitrary orientations of the “straight line” quiver. Stability is between the previously discussed weighted interleaving metric and the A-R quiver bottleneck metric—a flexible device proposed in 2014 by Emerson Escolar and Yasuaki Hiraoka.