String Diagrams for Physical Duoidal Categories (project proposal)
The following is a proposal for an undergraduate project.
String diagrams for symmetric monoidal categories are an excellent syntax: they capture all of the axioms of symmetric monoidal categories into a structure that requires no further quotienting or reasoning steps. The free strict symmetric monoidal category is captured combinatorially precisely by string diagrams.
Many variants are possible, but only some of them seem to be as elegant as the original one. Among these, I conjecture that the case of physical duoidal categories is particularly nice: string diagrams for physical duoidal categories are a variant of string diagrams for symmetric monoidal categories where wires form themselves a poset. Physical duoidal categories may have applications in the semantics of concurrency. I recently prepared a preprint with the description of these physical duoidal string diagrams
Research proposal. To develop string diagrams for physical duoidal categories. To find examples of their use and to prove results internally to these string diagrams. To consider other variants (duoidal categories, duoidally-enriched monoidal categories, …) and to relate them to their combinatorial description. To propose programming language constructs that take semantics here.
References.