From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4508 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: symmetric monoidal traces Date: Thu, 21 Aug 2008 10:39:08 -0300 (ADT) Message-ID: <20080821133908.A644E5C2A8@chase.mathstat.dal.ca> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019989 13624 80.91.229.2 (29 Apr 2009 15:46:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:29 +0000 (UTC) To: categories@mta.ca (Categories List) Original-X-From: rrosebru@mta.ca Thu Aug 21 19:22:19 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 21 Aug 2008 19:22:19 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KWIW9-00008k-Ja for categories-list@mta.ca; Thu, 21 Aug 2008 19:20:33 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 43 Original-Lines: 96 Xref: news.gmane.org gmane.science.mathematics.categories:4508 Archived-At: As Phil Scott immediately pointed out to me, the proof of this folklore result appears in a 2000 slide by Masahito Hasegawa entitled "A short proof of the uniqueness of trace on tortile categories", see http://www.kurims.kyoto-u.ac.jp/~hassei/papers/canonicaltrace.gif and also on p.23 in a paper by the same author that is to appear in MSCS, entitled "On traced monoidal closed categories": http://www.kurims.kyoto-u.ac.jp/~hassei/papers/tmcc-revised16may08.pdf -- Peter Peter Selinger wrote: > > [ Reminder from moderator: Attachments are not suitable for transmission > and the one mentioned below has been deleted (happily Peter provides a > url). On a similar note, recall that html messages are not suitable > either, and will not be posted. Please send text only. Thanks. ] > > Dear Mike, > > I don't know whether the proof of this result is spelled out in the > literature (uniqueness of the trace for dualizable objects in a > symmetric monoidal category). However, I have seen the result itself > mentioned, and it follows straightforwardly from Joyal-Street-Verity's > INT construction. > > As you have already said, every traced symmetric monoidal category C > can be embedded in a compact closed category INT(C), in such a way > that the trace of C is mapped to the canonical trace of > INT(C). Further, every strong monoidal functor preserves dual objects, > so if some object X of C has a "canonical" trace coming from a dual > object, then this also gets mapped to the canonical trace of INT(C). > Finally, since the functor is faithful, and maps the "given" and the > "canonical" trace to the same thing, it follows that the two traces > already coincide in C. The same argument works for balanced monoidal > categories. > > One can easily turn this argument into an elementary algebraic > proof. For an object X equipped with two traces Tr and Tr', consider > the following "interchange property" for f: A*X*X -> B*X*X: > > Tr'_X(Tr_X(f o (A*c))) = Tr_X(Tr'_X((B*c) o f)) > > See Figure (a) in the attached file for an illustration of this > property. It is akin to "symmetry sliding", except that it uses two > different traces. > > Figure (b) proves that if two traces satisfy the interchange property, > then they coincide. > > Finally, if one of the traces is the canonical one obtained from a > dual of X, then the interchange property holds by standard > diagrammatic reasoning (see Figure (c)). In particular, if a dual > exists, then the trace on X is unique. > > The proof, as shown in the attachment, is only correct in the > symmetric case. It also works in the balanced case, provided that one > inserts a twist map in the correct places. > > -- Peter > > (Attachment also available as > http://www.mathstat.dal.ca/~selinger/downloads/traces.gif) > > Michael Shulman wrote: > > > > Hi all, > > > > Can someone please point me to whatever categorical references exist > > regarding the canonical trace for dualizable objects in a symmetric > > monoidal category? I am particularly interested in (1) elementary > > expositions accessible to non-category-theorists and (2) any proofs of > > its uniqueness subject to various conditions. > > > > I know there are many references on traced monoidal categories, > > particularly with applications to computer science, but right now I am > > only interested in the symmetric monoidal trace. I also know that > > there are various reinventions/expositions of the notion in, for > > example, the topological literature (e.g. Dold-Puppe), but I would > > like an exposition not tied to any particular application. Finally, I > > know that the Joyal-Street-Verity paper "Traced Monoidal Categories" > > proves that the canonical symmetric (or, more precisely, balanced) > > monoidal trace is "universal" in that any traced monoidal category can > > be embedded in one equipped with the canonical trace, but as far as I > > can tell this need not determine the canonical trace uniquely. > > > > Thanks!! > > Mike > >