Daniel Cicala’s research while affiliated with University of California, Riverside and other places

To foster the study of networks on an abstract level, we further study the formalism of structured cospans. We define a topos of structured cospans and establish its theory of rewriting. For the rewrite relation, we propose a double categorical semantics to encode the compositionality of the structure cospans. For an application, we generalize the inductive viewpoint of graph rewriting to rewriting in a wider class of topoi.

The concept of a system has proliferated through natural and social sciences. While myriad theories of systems exist, there is no mathematical general theory of systems. In this thesis, we take a first step towards formulating such a theory. Our focus is on developing a syntax for compositional systems equipped with a rewriting theory. We pull from category theory and linguistics to accomplish this. The basic syntactical unit is a structured cospan and rewriting is introduced via the double pushout method. Two versions of rewriting are proposed: one that tracks intermediate steps and another disregards them. Benefits and drawbacks of both versions are discussed. We apply our results to the decomposition of closed systems, obtaining a structurally inductive viewpoint of rewriting such systems.

If $\mathbf{C}$ is a category with chosen pullbacks and a terminal object then, using a result of Shulman, we obtain a fully dualizable and symmetric monoidal bicategory $\mathbf{Sp(Sp(C))}$ whose objects are those of $\mathbf{C}$, whose morphisms are spans in $\mathbf{C}$, and whose 2-morphisms are isomorphism classes of spans of spans in $\mathbf{C}$. If $\mathbf{C}$ is a topos, the first author has previously constructed a bicategory $\mathbf{MonicSp(Csp(C))}$ whose objects are those of $\mathbf{C}$, whose morphisms are cospans in $\mathbf{C}$, and whose 2-morphisms are isomorphism classes of spans of cospans in $\mathbf{C}$ with monic legs. We prove this bicategory is also symmetric monoidal and even compact closed. We discuss applications of such bicategories to graph rewriting as well as to Morton and Vicary's combinatorial approach to Khovanov's categorified Heisenberg algebra.

We build a symmetric monoidal and compact closed bicategory by combining spans and cospans inside a topos. This can be used as a framework in which to study open networks and diagrammatic languages. We illustrate this framework with Coecke and Duncan's zx-calculus by constructing a bicategory with the natural numbers for 0-cells, the zx-calculus diagrams for 1-cells, and rewrite rules for 2-cells.

This paper presents a symmetric monoidal and compact closed bicategory that categorifies the zx-calculus developed by Coecke and Duncan. The 1-cells in this bicategory are certain graph morphisms that correspond to the string diagrams of the zx-calculus, while the 2-cells are rewrite rules.

We introduce the notion of a span of cospans and define, for them, horizonal and vertical composition. These compositions satisfy the interchange law if working in a topos $\mathbf{C}$ and if the span legs are monic. A bicategory is then constructed from $\mathbf{C}$-objects, $\mathbf{C}$-cospans, and doubly monic spans of $\mathbf{C}$-cospans. The primary motivation for this construction is an application to graph rewriting.