Fabio Zanasi

Interacting Hopf Algebras - the theory of linear systems

We present by generators and equations the theory IH whose free model is the category of linear 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 categorical quantum mechanics, concurrency and control theory. The equations of IH arise by distributive laws between PROPs of Hopf algebras – from which the name "Interacting Hopf algebras". The characterisation in terms of subspaces allows to think of IH as a string diagrammatic syntax for linear algebra: linear maps, spaces and their transformations are all faithfully represented in the graphical language, resulting in an alternative, often insightful perspective on the subject matter. As main application, we use IH to axiomatise a formal semantics of signal processing circuits, for which we study full abstraction and realisability. Our analysis suggests a reflection about the role of causality in the semantics of computing devices.

This is a joint work with Filippo Bonchi and Pawel Sobocinski.