From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6804 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Paper on "Partially traced categories" Date: Wed, 20 Jul 2011 12:32:59 -0300 (ADT) Message-ID: Reply-To: selinger@mathstat.dal.ca (Peter Selinger) NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1311177843 10202 80.91.229.12 (20 Jul 2011 16:04:03 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 20 Jul 2011 16:04:03 +0000 (UTC) To: categories@mta.ca (Categories List) Original-X-From: majordomo@mlist.mta.ca Wed Jul 20 18:03:59 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QjZFb-0005SG-O6 for gsmc-categories@m.gmane.org; Wed, 20 Jul 2011 18:03:55 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:33176) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QjZDj-000793-AG; Wed, 20 Jul 2011 13:01:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QjZDi-0003Lz-JW for categories-list@mlist.mta.ca; Wed, 20 Jul 2011 13:01:58 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6804 Archived-At: 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 ---------------------------------------------------------------------- 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. ---------------------------------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]