From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8205 Path: news.gmane.org!not-for-mail From: Tom Hirschowitz Newsgroups: gmane.science.mathematics.categories Subject: Re: generalised cartesian multicategories Date: Tue, 08 Jul 2014 09:06:14 +0200 Message-ID: References: <87lhs92nwb.fsf@hirscho.lama.univ-savoie.fr> Reply-To: Tom Hirschowitz 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 1404993993 13644 80.91.229.3 (10 Jul 2014 12:06:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 10 Jul 2014 12:06:33 +0000 (UTC) Cc: categories@mta.ca To: Michael Shulman Original-X-From: majordomo@mlist.mta.ca Thu Jul 10 14:06:27 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 1X5D7K-0000nA-8D for gsmc-categories@m.gmane.org; Thu, 10 Jul 2014 14:06:26 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45850) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X5D6d-000821-FG; Thu, 10 Jul 2014 09:05:43 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X5D6b-0006ni-SD for categories-list@mlist.mta.ca; Thu, 10 Jul 2014 09:05:41 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8205 Archived-At: Dear Mike,=20 Thanks a lot for your answer and hint, I'll try to figure this out. The answer to your question > Can you say anything about what it means for "cartesian > multicategories" to "make sense" for a monad T? is: not yet in general. 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 edge= s 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 of = an action E =E2=88=98 M =E2=86=92 E satisfying some axioms to be made precise,= e.g.,=20 (E M M =E2=86=92 E M =E2=86=92 E) =3D (E M M =E2=86=92 E M =E2=86=92 E), (E =E2=86=92 E M =E2=86=92 E) =3D id_E (E E M =E2=86=92 E M =E2=86=92 E) =3D (E E M =E2=86=92 E E =E2=86=92 E) (maybe more?).=20 Concretely, the domain of a morphism in such a T-multicategory is a finite sequence of paths in the underlying graph G, i.e., (ignoring the case of empty paths) tuples of tuples ((e=C2=B9=E2=82=81,=E2=80=A6,e=C2=B9=E2=82=99=E2=82=81), =E2=80=A6, (e=E1=B5=96=E2=82=81,=E2=80=A6,e=E1=B5=96=E2=82=99=E2=82=9A)), where target(e=E2=81=B1=E2=B1=BC) =3D source(e=E2=81=B1_{j+1}) (but not, e.= g., target(e=E2=81=B1=E2=82=99=E1=B5=A2) =3D source(e^{i+1}=E2=82=81) in general). Does that make any sense? Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]