2016/08/04 by Slavnov, Sergey
#Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO)
paper · doi:10.48550/arxiv.1608.01560
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.