From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/37 Path: news.gmane.org!not-for-mail From: street@ics.mq.edu.au Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on "On Closed Categories of Functors" Date: Mon, 2 Feb 2009 18:53:01 +1100 (EST) Message-ID: Reply-To: street@ics.mq.edu.au NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1233612375 26823 80.91.229.12 (2 Feb 2009 22:06:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 2 Feb 2009 22:06:15 +0000 (UTC) To: "Tony Meman" , "categories" Original-X-From: categories@mta.ca Mon Feb 02 23:07:30 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LU6wk-00009s-Kc for gsmc-categories@m.gmane.org; Mon, 02 Feb 2009 23:07:14 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LU6OH-0000zG-Ju for categories-list@mta.ca; Mon, 02 Feb 2009 17:31:37 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:37 Archived-At: > Let V be a symmetric monoidal closed category and C a small V-category. > Brian Day constructs a symmetric monoidal closed structure ([C,V],@,E) = on > the V-category of V-functors [C,V] for some cases [3.3, 3.6], e.g. if > (C,*,e) is a symmetric monoidal V-category [4.1]. The underlying > *category* > [C,V] of V-functors admits a closed symmetric monoidal structure from t= he > enriched one by taking the underlying functor of each V-functor, the > underlying natural transformation of each V-natural transformation. > > Because a closed symmetric monoidal category is canonically enriched ov= er > itself, the category [C,V] gets a [C,V] enrichment in this way. > > My question is: What does this [C,V]-enrichment of [C,V] have to do wit= h > the V-enrichment of [C,V]? > Suppose C have a terminal object t. One gets a evaluation functor > Ev_t:[C,V]-CAT-->V-CAT. Is this the connection between the two enrichme= nts? I think what you want here is the following observation. Every closed monoidal V-category E is also an E-category. The unit object j for tensor in E is a monoid and so E(j,-) : E --> V is a monoidal V-functor. Therefore by applying it on hom objects, it induces a 2-functor E-Cat --> V-Cat. In particular, you can apply the 2-functor to E itself to see it as a V-category. Your example is for E =3D [C,V]. Ross PS I have ordinary- (not enriched-) mailed your message to Brian himself. He may want to add something when he gets it. But I hope I have the story you need!