On partial traces and compactification of $\ast$-autonomous Mix-categories

We study the question when a $\ast$-autonomous Mix-category has a representation as a $\ast$-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 $\mathbf{K}$ has a mixed trace, then we can construct a compact category and structure preserving embedding of $\mathbf{K}$ 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 $\mathbf{K}$, which is necessary and sufficient for a structure preserving embedding of $\mathbf{K}$ into a compact one to exist. When this condition is satisfied, we construct a “free” or “minimal” mixed trace on $\mathbf{K}$ directly from the Mix-category structure, which gives us also a “free” compactification of $\mathbf{K}$.