Comb Diagrams for Discrete-Time Feedback

TL;DR

This paper formalizes the concept of comb diagrams with holes in symmetric monoidal categories, extending to infinite circuits, and explores their structure for applications in applied category theory.

Contribution

It introduces a formalization of diagrams with holes using dinaturality and coends, extending to infinite circuits with feedback in symmetric monoidal categories.

Findings

Infinite combs form symmetric monoidal categories.

Framework allows modeling of delay and feedback.

Preliminary work with potential for broad applications.

Abstract

The data for many useful bidirectional constructions in applied category theory (optics, learners, games, quantum combs) can be expressed in terms of diagrams containing "holes" or "incomplete parts", sometimes known as comb diagrams. We give a possible formalization of what these circuits with incomplete parts represent in terms of symmetric monoidal categories, using the dinaturality equivalence relations arising from a coend. Our main idea is to extend this formal description to allow for infinite circuits with holes indexed by the natural numbers. We show how infinite combs over an arbitrary symmetric monoidal category form again a symmetric monoidal category where notions of delay and feedback can be considered. The constructions presented here are still preliminary work.