Paper on "Partially traced categories"

[selinger@mathstat.dal.ca (Peter Selinger)]

Dear colleagues,

we just put a new paper on the ArXiv about partially traced
categories. They are almost the same thing as traced monoidal
categories (a la Joyal, Street, and Verity 1996), except that the
trace is a partially defined operation, subject to some axioms. The
main result is a representation theorem: every partially traced
category can be faithfully embedded in a totally traced category (and
conversely, every symmetric monoidal subcategory of a totally traced
category is partially traced; thus this representation theorem
completely characterizes partially traced categories).

Interestingly, there are some naturally occuring examples of partially
traced categories (such as on the category of vector spaces with
direct sum as the monoidal structure) that do not appear to be
embedded in any *naturally occuring* totally traced category (i.e.,
other than the one constructed by the completeness theorem).

The details of the paper appear below.

Best wishes, -- Octavio, Phil, and Peter

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 		     Partially traced categories

 	  Octavio Malherbe, Philip J. Scott, Peter Selinger

 		    http://arxiv.org/abs/1107.3608

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.

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