From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/764 Path: news.gmane.org!not-for-mail From: Max Kelly Newsgroups: gmane.science.mathematics.categories Subject: Re: Units in lax and oplax monoidal functors Date: Mon, 22 Jun 1998 13:29:54 +0200 (MET DST) Message-ID: <199806221129.NAA20705@ifdh2.sc.ucl.ac.be> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241017168 27268 80.91.229.2 (29 Apr 2009 14:59:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:59:28 +0000 (UTC) To: cat-dist@mta.ca, dyetter@math.ksu.edu Original-X-From: cat-dist Mon Jun 22 14:18:49 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA32428 for categories-list; Mon, 22 Jun 1998 12:52:59 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from ifdh2.sc.ucl.ac.be (ifdh2.sc.ucl.ac.be [130.104.10.106]) by mailserv.mta.ca (8.8.8/8.8.8) with ESMTP id IAA10710 for ; Mon, 22 Jun 1998 08:29:58 -0300 (ADT) X-Received: (from kelly_m@localhost) by ifdh2.sc.ucl.ac.be (8.8.8/dvg-ifdh2) id NAA20705; Mon, 22 Jun 1998 13:29:54 +0200 (MET DST) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 18 Xref: news.gmane.org gmane.science.mathematics.categories:764 Archived-At: David Yetter asks about coherence results for monoidal functors. These were studied in the PhD thesis of my then-student Geoff Lewis round about 1971, and he has an article about them in that Springer Lecture Notes volume - was it number 129 ? - on coherence in categories, edited by Saunders Mac Lane in the early 1970s. It is one of the cases covered by the "club" idea, where the free structure on 1 tells you all about the free structure on any category. Moreover the case of two monoidal categories and a monoidal functor (lax, of course) is interesting in that Lewis finds the club COMPLETELY, even though it is false that "every diagram commutes". What is true, if f is the monoidal functor, is that a diagram commutes if its codomain has the form f(x), in contrast to say f(x)of(y) where o is the tensor product. Lewis also studies there the case of a monoidal f between monoidal CLOSED categories (everything symmetric), getting for these a PARTIAL determination of the club, like that of Kelly and Mac Lane for a single symmetric monoidal category. Max Kelly.