2015/01/01 by Fabio Zanasi, Zanasi, Fabio, Zanasi Fabio · 2 citations
Computer Science · #Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Formal Methods in Verification #Logic in Computer Science (cs.LO) #Logic, programming, and type systems #Model-Driven Software Engineering Techniques #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1805.03032
openalex publication_date 2015/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/14
We present by generators and equations the algebraic theory IH whose free model is the category oflinear subspaces over a field k. Terms of IH are string diagrams which, for different choices of k, express different kinds of networks and graphical formalisms used by scientists in various fields, such as quantumcircuits, electrical circuits and Petri nets. The equations of IH arise by distributive laws between Hopfalgebras - from which the name interacting Hopf algebras. The characterisation in terms of subspacesallows to think of IH as a string diagrammatic syntax for linear algebra: linear maps, spaces and theirtransformations are all faithfully represented in the graphical language, resulting in an alternative, ofteninsightful perspective on the subject matter. As main application, we use IH to axiomatise a formalsemantics of signal processing circuits, for which we study full abstraction and realisability. Our analysissuggests a reflection about the role of causality in the semantics of computing devices.