From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8209 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: generalised cartesian multicategories Date: Wed, 9 Jul 2014 16:00:58 -0700 Message-ID: References: <87lhs92nwb.fsf@hirscho.lama.univ-savoie.fr> 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 1404995183 29239 80.91.229.3 (10 Jul 2014 12:26:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 10 Jul 2014 12:26:23 +0000 (UTC) Cc: "categories@mta.ca" To: Tom Hirschowitz Original-X-From: majordomo@mlist.mta.ca Thu Jul 10 14:26:18 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 1X5DQX-0001bD-6U for gsmc-categories@m.gmane.org; Thu, 10 Jul 2014 14:26:17 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45880) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X5DQ1-0000j5-Gn; Thu, 10 Jul 2014 09:25:45 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X5DPz-00072Z-JN for categories-list@mlist.mta.ca; Thu, 10 Jul 2014 09:25:43 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8209 Archived-At: Ok, here's a more precise version of my guess. On Tue, Jul 8, 2014 at 12:06 AM, Tom Hirschowitz wrote: > But i can say what it'd like it to mean for my > particular monad T =3D fm fc: for any graph G, consider the span fm(G) -|= =E2=86=92 > fm(G) defined by > > =E2=88=91=E2=82=98 G=E1=B5=90 =E2=86=90 =E2=88=91_{m,n} m=E2=81=BF =E2=8B= =85 G=E1=B5=90 =E2=86=92 =E2=88=91=E2=82=99 G=E2=81=BF > (m,e) =E2=86=A4 (m,n,f,e) =E2=86=A6 (n, e =E2=88=98 f) (both on ed= ges and vertices). > > If i'm correct, this forms a monad in Span(Gph), by composing underlying > maps (f here), say M. > > Cartesian structure on a T-multicategory E : TG -|=E2=86=92 G consists o= f an > action E =E2=88=98 M =E2=86=92 E satisfying some axioms to be made precis= e, e.g., Tom clarified by private email that if the definition of M from G is denoted M_G, then in the last paragraph above he means M_{fc(G)}, so that M : TG -|=E2=86=92 TG and hence E =E2=88=98 M : TG -|=E2=86=92 G. A monad in a bicategory of spans is, of course, an internal category. I suspect that your construction G |=E2=86=92 M_{fc(G)} can be extended to = a monad on the bicategory of internal profunctors in Gph, and that your cartesian T-multicategories are generalized multicategories for this monad (which are "object-discrete" in the sense of my paper with Geoff that I cited in my last email, since their underlying object is a graph G rather than an internal category in graphs). Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]