From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7093 Path: news.gmane.org!not-for-mail From: "Szlachanyi Kornel" Newsgroups: gmane.science.mathematics.categories Subject: skew-monoidal category? Date: Fri, 2 Dec 2011 11:43:08 +0100 Message-ID: Reply-To: "Szlachanyi Kornel" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-2 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1322833471 17322 80.91.229.12 (2 Dec 2011 13:44:31 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 2 Dec 2011 13:44:31 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Dec 02 14:44:27 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 1RWTPe-0004ga-Me for gsmc-categories@m.gmane.org; Fri, 02 Dec 2011 14:44:26 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:33021) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RWTOd-0005GT-3c; Fri, 02 Dec 2011 09:43:23 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RWTOb-0003Gb-FR for categories-list@mlist.mta.ca; Fri, 02 Dec 2011 09:43:21 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7093 Archived-At: 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/ ]