Title: A Topological Perspective on Interacting Algebraic Theories,
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.