The compact double category $\mathbf{Int}(\mathbf{Poly}_*)$ models control flow and data transformations

Abstract

Hasegawa showed that control flow in programming languages -- while loops and if-then-else statements -- can be modeled using traced cocartesian categories, such as the category $\mathbf{Set}_*$ of pointed sets. In this paper we define an operad $\mathscr{W}$ of wiring diagrams that provides syntax for categories whose control flow moreover includes data transformations, including deleting, duplicating, permuting, and applying pre-specified functions to variables. In the most basic version, the operad underlies $\mathbf{Int}(\mathbf{Poly}_*)$, where $\mathbf{Int}(\mathscr{T})$ denotes the free compact category on a traced category $\mathscr{T}$, as defined by Joyal, Street, and Verity; to do so, we show that $\mathbf{Poly}_*$, as well as any multivariate version of it, is traced. We show moreover that whenever $\mathscr{T}$ is uniform -- a condition also defined by Hasegawa and satisfied by $\mathbf{Int}(\mathscr{T})$ -- the resulting $\mathbf{Int}$-construction extends to a double category $\mathbb{I}\mathbf{nt}(\mathscr{T})$, which is compact in the sense of Patterson. Finally, we define a universal property of the double category $\mathbb{I}\mathbf{nt}(\mathbf{Poly}_*)$ and $\mathbb{I}\mathbf{nt}(\mathbf{Set}_*)$ by which one can track trajectories as they move through the control flow associated to a wiring diagram.