String diagrams for traced and compact categories are oriented 1-cobordisms

TL;DR

This paper introduces a new perspective on string diagrams by modeling them as oriented cobordisms, establishing an equivalence with traced and compact categories, and providing a characterization of the 2-category of traced categories.

Contribution

It presents an operadic framework that encodes traced and compact categories as labeled cobordisms, offering a new geometric interpretation and a characterization of their 2-categories.

Findings

String diagrams are modeled as labeled 1-cobordisms.

Equivalence established between cobordism algebras and traced/compact categories.

Provides a geometric characterization of the 2-category of traced categories.

Abstract

We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob/O, where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully encoded by Cob/O in the sense that there is an equivalence between (Cob/O)-algebras, for varying O, and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O. As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms of those of monoidal and compact categories, without any reference to the usual structures or axioms of traced categories. In an appendix we offer a complete proof of the well-known relationship between the 2-category of monoidal categories with strong monoidal functors and the 2-category of monoidal categories whose object set is free with strict functors; similarly for traced and compact categories.