{"ID":2874356,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.05462","arxiv_id":"2509.05462","title":"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.","short_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...","url_abs":"https://arxiv.org/abs/2509.05462","url_pdf":"https://arxiv.org/pdf/2509.05462v1","authors":"[\"Grigory Kondyrev\",\"David I. Spivak\"]","published":"2025-09-05T19:35:08Z","proceeding":"math.CT","tasks":"[\"math.CT\",\"cs.PL\"]","methods":"[]","has_code":false}