From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8203 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: generalised cartesian multicategories Date: Mon, 7 Jul 2014 20:07:22 -0700 Message-ID: Reply-To: Michael Shulman NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1404802595 16762 80.91.229.3 (8 Jul 2014 06:56:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 8 Jul 2014 06:56:35 +0000 (UTC) Cc: categories@mta.ca To: Tom Hirschowitz Original-X-From: majordomo@mlist.mta.ca Tue Jul 08 08:56:29 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1X4PKC-0000qf-9k for gsmc-categories@m.gmane.org; Tue, 08 Jul 2014 08:56:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45399) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X4PJk-0008Sp-UQ; Tue, 08 Jul 2014 03:55:56 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X4PJm-0000tD-BS for categories-list@mlist.mta.ca; Tue, 08 Jul 2014 03:55:58 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8203 Archived-At: Can you say anything about what it means for "cartesian multicategories" to "make sense" for a monad T? There is a more general notion of generalized multicategory which takes place in a more general bicategory (or, better, a double category) than T-spans, and which includes cartesian multicategories as a special case (see http://tac.mta.ca/tac/volumes/24/21/24-21abs.html for a unified account, as well as references to a lot of prior work). I suspect that your "cartesian T-multicategories" are probably generalized multicategories in this sense relative to some other monad built out of T. Mike On Fri, Jul 4, 2014 at 7:34 AM, Tom Hirschowitz wrote: > > Dear all, > > Cartesian multicategories are multicategories equipped with > `contraction' and `weakening' operations. E.g., contraction associates > to any morphism x=E2=82=81, =E2=80=A6, x=E2=82=99 =E2=86=92 y and 1 =E2= =89=A4 i =E2=89=A4 n such that x=E2=B1=BC =3D x_{j+1} for > some j a morphism x=E2=82=81, =E2=80=A6, x=E2=B1=BC, x_{j+2}, =E2=80=A6 x= =E2=82=99 =E2=86=92 y. > > On the other hand we have generalised multicategories, which are monads > in the bicategory of T-spans, for some cartesian monad T. > > I'm currently considering such a monad T for which cartesian > multicategories make obvious sense, and wonder whether anyone has worked > out a general setting for this. I.e., are there some known conditions on > the monad T for cartesian T-multicategories to make sense? Of > particular interest would be a setting in which free cartesian > T-multcategories exist (over T-graphs). > > For those interested, the monad in question is on graphs. It's the > composite of > > - the `free category' monad fc, and > > - the `free monoidal graph' monad fm, mapping any graph s,t : E =E2=86= =92 T to > s*,t* : E* =E2=86=92 T*, > > made into a monad via the obvious distributive law > > fc =E2=88=98 fm =E2=86=92 fm =E2=88=98 fc. > > Any hints? > Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]