From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4200 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: question about monoidal categories Date: Thu, 7 Feb 2008 22:36:24 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019789 12187 80.91.229.2 (29 Apr 2009 15:43:09 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:43:09 +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 1JNYMo-0004Ix-RO for categories-list@mta.ca; Fri, 08 Feb 2008 14:54:30 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 22 Original-Lines: 96 Xref: news.gmane.org gmane.science.mathematics.categories:4200 Archived-At: Dear Paul, Here is one possible answer to your question. One has a notion of action of a monoidal category V on an arbitrary category X, which generalises that of action by a monoid on a set; thus we have a functor (-) * (-): X x V --> X together with natural isomorphisms x * I ~ x and x * (v * w) ~ (x * v) * w satisfying pentagon and triangle axioms. Formally, one may define an action of V on X to be a strong monoidal functor V --> [X, X], where the latter is equipped with its compositional monoidal structure. If we are given two categories X and Y equipped with an action by V, then we have a notion of equivariant morphism between them; namely a functor F: X --> Y together with natural morphisms m_{x, v} : F(x) * v --> F(x * v) obeying axioms like those for a monoidal functor. This is what one might call a lax equivariant morphism; if the m_{x, v}'s are all invertible we should rather call it strong, whilst if they point in the opposite direction then what we have is an oplax morphism. The situation you have described is a special case of an lax equivariant morphism. You have a monoidal category M, and a functor F : M --> S. Now, M has a canonical action on itself induced by tensoring on the right (the "right regular representation"); and it has a trivial action on S given by s * m = s for all s and m. Your natural transformation beta can now be written as beta(p, a) : F(p) * a --> F(p * a), and your two axioms are precisely the axioms required for beta to equip F with the structure of a lax equivariant morphism. This whole area of monoidal actions is slightly folklorish but a useful source is: George Janelidze and Max Kelly, "A note on actions of a monoidal category", TAC Vol. 9, No. 4 Also worth mentioning is the work of Paddy McCrudden who has studied actions by a symmetric monoidal V under the name "V-actegories". Hope this is of some help, Richard --On 07 February 2008 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 > > >