Partially traced categories

TL;DR

This paper provides a representation theorem for partially traced categories, showing they can be faithfully embedded in totally traced categories, and characterizes them via symmetric monoidal subcategories.

Contribution

It introduces a representation theorem for partially traced categories using Freyd's paracategories and a partial Int-construction, offering a complete characterization.

Findings

Every partially traced category embeds into a totally traced category.

Symmetric monoidal subcategories of totally traced categories are partially traced.

The main technique involves Freyd's paracategories and a partial Int-construction.

Abstract

This paper deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every symmetric monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int-construction.