Protocol Choice and Iteration for the Free Cornering
Chad Nester, Niels Voorneveld
TL;DR
The paper advances the categorical modeling of concurrent interaction by extending the free cornering of a symmetric monoidal category to support branching and iterative communication protocols. It develops the free cornering with choice and the free cornering with iteration, proving that each inherits the monoidal double-category structure and extends crossing-cell theory accordingly. A central contribution is the construction of a double category of stateful transformations that models these extended cornerings, providing concrete realizations via strong functors and strong natural transformations, and connecting to computational effects. The results refine the connection to session-type style protocols, enabling richer interaction patterns, and open avenues for further exploration of active iteration, term calculi, and denotational/operational semantics within this double-category framework.
Abstract
We extend the free cornering of a symmetric monoidal category, a double categorical model of concurrent interaction, to support branching communication protocols and iterated communication protocols. We validate our constructions by showing that they inherit significant categorical structure from the free cornering, including that they form monoidal double categories. We also establish some elementary properties of the novel structure they contain. Further, we give a model of the free cornering in terms of strong functors and strong natural transformations, inspired by the literature on computational effects.