From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3826 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Actions of monoidal functors [was Re: Arens product] Date: Tue, 17 Jul 2007 14:11:46 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241019547 10473 80.91.229.2 (29 Apr 2009 15:39:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:07 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Jul 17 21:30:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 17 Jul 2007 21:30:43 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IAxKz-0005Hf-Qb for categories-list@mta.ca; Tue, 17 Jul 2007 21:24:17 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 20 Original-Lines: 45 Xref: news.gmane.org gmane.science.mathematics.categories:3826 Archived-At: 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.