From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7094 Path: news.gmane.org!not-for-mail From: Robin Houston Newsgroups: gmane.science.mathematics.categories Subject: Re: skew-monoidal category? Date: Fri, 2 Dec 2011 14:23:20 +0000 Message-ID: References: Reply-To: Robin Houston NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1322947700 11753 80.91.229.12 (3 Dec 2011 21:28:20 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 3 Dec 2011 21:28:20 +0000 (UTC) Cc: categories@mta.ca To: Szlachanyi Kornel Original-X-From: majordomo@mlist.mta.ca Sat Dec 03 22:28:15 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RWx82-00073M-Rx for gsmc-categories@m.gmane.org; Sat, 03 Dec 2011 22:28:15 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:43205) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RWx5U-0003bn-OR; Sat, 03 Dec 2011 17:25:36 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RWx5T-0000su-7I for categories-list@mlist.mta.ca; Sat, 03 Dec 2011 17:25:35 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7094 Archived-At: I don=92t know the full history of this idea; I came across it in the work = of Marco Grandis on directed homotopy, such as http://www.dima.unige.it/~grandis/LCat.pdf Grandis considers a generalisation of bicategories, rather than monoidal categories, but of course the monoidal version is the special case of a bicategory with a single object. Actually Grandis has the maps you call eta and eps going the other way, so perhaps the precise notion you=92re describing is more closely related to Burroni=92s 1971 notion of =91pseudocategory=92. Anyway I expect some of the more knowledgeable members of this list will be able to give you a better answer! All the best, Robin 2011/12/2 Szlachanyi Kornel > Dear All, > > I wonder if the following notion has already a name and disscussed > somewhere: It is like a monoidal category but the associator and units > are not invertible. (Lax monoidal categories share this property but they > seem to treat the units differently.) It has left and right versions, the > "right-monoidal" category consists of > > a category C, > a functor C x C --> C, |--> M*N, > an object R > and natural transformations > gamma_L,M,N: L*(M*N) --> (L*M)*N > eta_M: M --> R*M > eps_M: M*R -->M > > satisfying 5 axioms (1 pentagon, 3 triangles and eps_R o eta_R =3D R) tha= t > are obtained from the usual monoidal category axioms by expressing > everything in terms of the associator, the right unit (eps), and the > inverse left unit (eta) never using their inverses. > > I find this structure interesting because of the following: > > Thm: Let R be a ring. Closed right-monoidal structures on the category M_= R > of right R-modules are (up to approp. isomorphisms on both sides) precise= ly > the right R-bialgebroids. > > (The ordinary monoidal structure remains hidden in the special nature of > M_R.) > > I would thank for any suggestion. > > Kornel Szlachanyi > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]