A Topological Perspective on Interacting Algebraic Theories

TL;DR

This paper introduces a topological framework for analyzing higher algebraic theories using concepts from higher categories, string diagrams, and directed topology, with applications to quantum computation and the ZX calculus.

Contribution

It develops a new language of directed topology for the compositional study of algebraic theories, connecting computads with topological notions and providing a unified framework.

Findings

Constructs computads with topological analogues for algebraic theories.

Captures homomorphisms and actions within a topological framework.

Reconstructs a fragment of the ZX calculus using the developed methods.

Abstract

Techniques from higher categories and higher-dimensional rewriting are becoming increasingly important for understanding the finer, computational properties of higher algebraic theories that arise, among other fields, in quantum computation. These theories have often the property of containing simpler sub-theories, whose interaction is regulated in a limited number of ways, which reveals a topological substrate when pictured by string diagrams. By exploring the double nature of computads as presentations of higher algebraic theories, and combinatorial descriptions of "directed spaces", we develop a basic language of directed topology for the compositional study of algebraic theories. We present constructions of computads, all with clear analogues in standard topology, that capture in great generality such notions as homomorphisms and actions, and the interactions of monoids and comonoids that lead to the theory of Frobenius algebras and of bialgebras. After a number of examples, we describe how a fragment of the ZX calculus can be reconstructed in this framework.