Compactification of *-autonomous categories

TL;DR

This paper investigates conditions under which *-autonomous categories can be represented as compact *-autonomous categories, identifying torsion-free categories as key and linking them to canonical partial traces.

Contribution

It establishes necessary and sufficient conditions for *-autonomous categories to be represented as compact, introducing the concept of torsion-free categories and their properties.

Findings

Weak distributivity maps are monic or epic in torsion-free categories.

Torsion-free categories possess canonical partial traces.

Conditions for representation as compact categories are characterized by monic or epic maps.

Abstract

We study the question when a *-autonomous (Mix-)category has a representation as a $*$-autonomous category of a compact one. We prove that necessary and sufficient condition is that weak distributivity maps are monic (or, equivalently epic). For a Mix-category, this condition is, in turn, equivalent to the requirement that Mix-maps be monic (or epic). We call categories satisfying this property torsion-free. An important side result is that torsion-free categories have canonical partial traces.