From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7384 Path: news.gmane.org!not-for-mail From: Bruce Bartlett Newsgroups: gmane.science.mathematics.categories Subject: Re: Examples of symmetric monoidal bicategories Date: Mon, 16 Jul 2012 23:30:55 +0200 Message-ID: References: Reply-To: Bruce Bartlett NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1342561601 25216 80.91.229.3 (17 Jul 2012 21:46:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 17 Jul 2012 21:46:41 +0000 (UTC) To: roman.krenicky@cs.manchester.ac.uk, categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Jul 17 23:46:40 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1SrFbJ-0004LJ-Td for gsmc-categories@m.gmane.org; Tue, 17 Jul 2012 23:46:38 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:57267) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SrFav-0004p1-4x; Tue, 17 Jul 2012 18:46:13 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SrFaw-0000TP-GG for categories-list@mlist.mta.ca; Tue, 17 Jul 2012 18:46:14 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7384 Archived-At: Representations of 2-groups are nice examples of symmetric monoidal bicategories with non-trivial coherence data (the "15j" symbols which categorify the "6j symbols" from representation categories of groups). See eg. "2-group representations for spin foams", by Baratin and Wise for a pointer to further literature, http://arxiv.org/abs/0910.1542. Regards, Bruce Bartlett On Mon, Jul 16, 2012 at 6:49 AM, Steve Lack wrote: > > On 15/07/2012, at 4:11 AM, Mike Stay wrote: > >> On Thu, Jul 12, 2012 at 7:59 AM, Roman Krenick=FD >> wrote: >>> Dear all, >>> >>> I'm looking for examples of symmetric monoidal bicategories (where the >>> structure is genuinely weak, i.e. the various isomorphisms are not >>> identities) and I would appreciate some help. -- Actually, strict >>> 2-categories with (genuinely) weak monoidal structure would be even mo= re >>> interesting, but I found it almost impossible to find anything on that= . >>> >>> As I am using these categories as models, I need some structure that i= s >>> "concrete enough" to do calculations with (while being as simple as >>> possible). >>> >>> Right now I'm considering the bicategory of rings (or monoids or >>> fields), bimodules over them, and bimodule homomorphisms, where the >>> monoidal structure is defined by the tensor product etc. (Pointers to >>> detailed accounts of this category would be very much appreciated, too= . >>> I've only found fairly sketchy mentions in the literature.) >>> >>> I would be grateful about any other examples of this kind! >> >> Spans of sets. The cartesian product of sets has an associator, so >> the tensor product of spans does, too. >> > > That's a good example; it's also very similar to the example of modules. > > For any braided monoidal category V with coequalizers of reflexive pairs, > which are > preserved by tensoring on either side, there's a monoidal bicategory Mod(= V) > in which the objects are the monoids in V, the 1-cells from a monoid M to= a > monoid N are objects equipped with a left M-action and a right N-action > satisfying > the obvious compatibility condition, and the 2-cells are the morphisms in > M compatible > with the two actions. > > If you start with the monoidal category Ab of abelian groups, with the > usual tensor > product, you get the bicategory of rings, bimdules, and bimodule > homomorphisms. > > If you start with the *opposite category* of Set, with tensor product > given by the > cartesian product in Set (and so by the coproduct in Set^op), then a > monoid is just > a set, an object with compatible left and right actions is a Span, and a > 2-cell is a > morpihsm of spans. It then turns out that Mod(Set^op) is Span^co, where > the co > means that the direction of the 2-cells is reversed. > > You could also consider a larger monoidal bicategory Mod-V, where the > objects > are not just monoids in V but V-enriched categories; then the 1-cells wil= l > be enriched > profunctors. > > Steve Lack. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]