From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4197 Path: news.gmane.org!not-for-mail From: Sam Staton Newsgroups: gmane.science.mathematics.categories Subject: Re: question about monoidal categories Date: Fri, 8 Feb 2008 09:51:09 +0000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v753) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019787 12175 80.91.229.2 (29 Apr 2009 15:43:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:43:07 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Feb 8 15:00:05 2008 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 08 Feb 2008 15:00:05 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JNYPB-0004d5-BI for categories-list@mta.ca; Fri, 08 Feb 2008 14:56:57 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 19 Original-Lines: 67 Xref: news.gmane.org gmane.science.mathematics.categories:4197 Archived-At: Hello, here is an answer to Paul's question: The data (F,beta) defines a (lax?) functor between "monoidal categories with right units but not left units" [henceforth MR-categories]. Every pointed category can be considered as an MR-category, in fact a strict one. The tensor product is left projection. The right unit is the point of the category (although every object of the category behaves like a right unit for this category.) In particular, the category S in Paul's email, with point F(1), can be seen as an MR-category. Of course, every monoidal category, including M, is an MR-category too. The data for a lax MR-functor M->S consists of a functor F:M->S, a morphism F(i)->F(i), which we can take as identity, and a natural transformation F(p)*F(a) --> F(p*a) which in this case amounts to a map beta:F(p) --> F(p*a) A monoidal functor must satisfy three coherence conditions, for associativity, left identity and right identity. For an MR-functor, there are only axioms for associativity and right identity, and these are exactly the axioms that Paul gave. Hope that makes sense! All the best, Sam. On 7 Feb 2008, at 20:05, Paul B Levy wrote: > > > > Let F be a functor from a monoidal category M to a category S. > > We are given > > beta(p,a) : F(p) --> F(p*a) > > natural in p,a in M. > > If I tell you that, in addition to naturality, beta is "monoidal", I'm > sure you will immediately guess what I mean by this, viz. > > (a) for any p,a,b in M > > beta(p,a) ; beta(p*a,b) = beta(p,a*b) ; F(alpha(p,a,b)) > > (b) for any p in M > > beta(p,1) = F(rho(p)) > > Yet I cannot see any reason for giving the name "monoidality" to > (a)-(b). > > It doesn't appear to be a monoidal natural transformation in the > official > sense. There are no monoidal functors in sight. > > Can somebody please justify my usage? > > Paul > >