From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3829 Path: news.gmane.org!not-for-mail From: rjwood@mathstat.dal.ca (RJ Wood) Newsgroups: gmane.science.mathematics.categories Subject: Re: tensorial strength Date: Wed, 18 Jul 2007 12:23:56 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019548 10489 80.91.229.2 (29 Apr 2009 15:39:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:08 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Jul 18 14:29:08 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 18 Jul 2007 14:29:08 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IBDIQ-0001BL-9O for categories-list@mta.ca; Wed, 18 Jul 2007 14:26:42 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 23 Original-Lines: 75 Xref: news.gmane.org gmane.science.mathematics.categories:3829 Archived-At: (I wrote this to Jeff and Anders a few minutes ago. Since Anders has replied to all I am circulating it more widely.) Dear Jeff and Anders In my thesis (supervised by Bob Pare at Dalhousie in 1976) I considered (for reasons I won't labour here) (V^op,set)-categories, for monoidal V. The monoidal structure I took on (V^op,set) was Brian Day's convolution. Necessarily, a (V^op,set)-category A gives rise to a functor P:A^op x V^op x A ---> set and this reveals that there are three special kinds of (V^op,set)-categories: 1) those for which P(-,v,b) is representable, for all v and b, by {v,b} say 2) those for which P(a,-,b) is representable, for all a and b, by [a,b] say 3) those for which P(a,v,-) is representable, for all a and v, by v@a say. Ordinary V-categories are given by 2). The others have been known by various names but they are best understood in terms of actions. Now suppose that F:A--->B is a (V^op,set)-functor where A is of type i) and B is of type j) as above. Each of the nine possibilities admits a simple encoding of the enrichment as displayed in the following table: i)\j) 1) 2) 3) 1) F{v,b}--->{v,Fb} v--->[F{v,b},Fb] v@F{v,b}--->Fb 2) Fa--->{[a,b],Fb} [a,b]--->[Fa,Fb] [a,b]@Fa--->Fb 3) Fa--->{v,F(v@a)} v--->[Fa,F(v@a)] v@Fa--->F(v@a) Susan Niefield, Robin Cockett, and I are writing a paper whose sequel will deal with later developments of this topic. Best to all, Richard > Dear all, > > Anders Kock's reply to Yemon Choi gives me a good opportunity to pose a > question which I have been meaning to ask the list for a while: > > > The V-enrichment ("strength") of an endofunctor T on V can be encoded > > without reference to the closed structure of V as a transformation > > T(A)@B-->T(A@B) ("tensorial strength", introduced in [4]). > > This notion of "tensorial strength" is just a special case of what > I would call "an action of a monoidal functor on a (mere) functor". > Specifically, it is a right-action of the identity monoidal functor > on the functor T. > > In general, given a monoidal functor M:V-->W and a functor T:V-->W, a > right-action of M on T should be a n.t. of the form T(A)@M(B)-->T(A@B) > satisfying the obvious associativity and unitality axioms. > > For instance, if we regard a G-graded algebra as a monoidal functor G-->Vec, > then a right-action of this on a mere functor G-->Vec is precisely the same > thing as a G-graded right-module. [Here the monoid G (G can also stand for > grading-object!) is considered as a discrete monoidal category.] > > I have always assumed that this concept is well-known, but I haven't > succeeded in finding a reference in the literature for it... perhaps > some of the more well-read readers of this list could help me out? > > Cheers, > Jeff. > > P.S. Upon reviewing [4], I see that there is a more general notion of > tensorial strength which can be applied to a functor A-->B whenever A > and B are tensored over some monoidal category V. There is a similar > adaptation of the notion of action of a monoidal functor V-->W to > functors A-->B whenever A is tensored over (or I would say, acted on by) > V, and B over (by) W. > > > [4] Strong functors and monoidal monads, Archiv der Math. 23 (1972), > > 113-120.