From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4199 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: question about monoidal categories Date: Thu, 7 Feb 2008 18:58:47 -0500 (EST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019788 12183 80.91.229.2 (29 Apr 2009 15:43:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:43:08 +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 1JNYNc-0004QF-Vg for categories-list@mta.ca; Fri, 08 Feb 2008 14:55:21 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 21 Original-Lines: 68 Xref: news.gmane.org gmane.science.mathematics.categories:4199 Archived-At: Hi Paul,=20 I think that I have written previously to the list about the=20 possibility of a monoidal functor acting on a mere functor,=20 and what you have is an instance of this notion. =20 Here the monoidal functor is the unique functor M ---> T, where T is the terminal (monoidal) category. Your beta is=20 a right action of this guy on F. In general, a right action of monoidal A --U--> C on mere=20 P --F--> S requires a right action of A on P and a right=20 action of C on S as well as a natural transformation=20 F(p)*U(a) --beta(p,a)--> F(p*a)=20 satisfying the appropriate associativity and unit axioms. In your case S is equipped with the trivial right T-action=20 (x*1=3Dx), and M with its canonical right M-action (a*b=3Da*b). =20 The axioms are identical. Cheers, Jeff. --- Paul B Levy wrote: >=20 >=20 >=20 > Let F be a functor from a monoidal category M to a category S. >=20 > We are given >=20 > beta(p,a) : F(p) --> F(p*a) >=20 > natural in p,a in M. >=20 > 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. >=20 > (a) for any p,a,b in M >=20 > beta(p,a) ; beta(p*a,b) =3D beta(p,a*b) ; F(alpha(p,a,b)) >=20 > (b) for any p in M >=20 > beta(p,1) =3D F(rho(p)) >=20 > Yet I cannot see any reason for giving the name "monoidality" to (a)-(b= ). >=20 > It doesn't appear to be a monoidal natural transformation in the offici= al > sense. There are no monoidal functors in sight. >=20 > Can somebody please justify my usage? >=20 > Paul >=20 >=20 >=20 Be smarter than spam. See how smart SpamGuard is at giving junk ema= il the boot with the All-new Yahoo! Mail. Click on Options in Mail and s= witch to New Mail today or register for free at http://mail.yahoo.ca=20