On partial traces and compactification of $*$-autonomous Mix-categories

TL;DR

This paper investigates conditions under which a $*$-autonomous Mix-category can be embedded into a compact one, introducing a mixed trace concept that characterizes such embeddings and enables constructing minimal compactifications.

Contribution

It defines the mixed trace operation, establishes its equivalence with embeddings into compact categories, and provides a necessary and sufficient condition for such embeddings.

Findings

A mixed trace can be induced by a structure-preserving embedding into a compact category.

Existence of a mixed trace characterizes when a Mix-category can be embedded into a compact one.

A specific interaction condition of Mix- and coevaluation maps is necessary and sufficient for embedding.

Abstract

We study the question when a $*$-autonomous Mix-category has a representation as a $*$-autonomous Mix-subcategory of a compact one. We define certain partial trace-like operation on morphisms of a Mix-category, which we call a mixed trace, and show that any structure preserving embedding of a Mix-category into a compact one induces a mixed trace on the former. We also show that, conversely, if a Mix-category ${\bf K}$ has a mixed trace, then we can construct a compact category and structure preserving embedding of ${\bf K}$ into it, which induces the same mixed trace. Finally, we find a specific condition expressed in terms of interaction of Mix- and coevaluation maps on a Mix-category ${\bf K}$, which is necessary and sufficient for a structure preserving embedding of ${\bf K}$ into a compact one to exist. When this condition is satisfied, we construct a "free" or "minimal" mixed trace on ${\bf K}$ directly from the Mix-category structure, which gives us also a "free" compactification of ${\bf K}$.