From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/35 Path: news.gmane.org!not-for-mail From: Tony Meman Newsgroups: gmane.science.mathematics.categories Subject: Question on "On Closed Categories of Functors" Date: Sun, 1 Feb 2009 20:11:25 +0100 Message-ID: Reply-To: Tony Meman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1233532811 19008 80.91.229.12 (2 Feb 2009 00:00:11 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 2 Feb 2009 00:00:11 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Mon Feb 02 01:01:26 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 1LTmFg-0003IO-6n for gsmc-categories@m.gmane.org; Mon, 02 Feb 2009 01:01:24 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LTllR-0001i6-Pf for categories-list@mta.ca; Sun, 01 Feb 2009 19:30:09 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:35 Archived-At: Dear category theorists, I have a question concerning the paper "On Closed Categories of Functors" from Brian Day (By the way, this is an excellent paper). Let V be a symmetric monoidal closed category and C a small V-category. The (ordinary) category [C,V] of V-functors admits the sturcure of a V-category in a canonical way. A symmetric monoidal V-category is the enriched analogue of a symmetric-monoidal structure on an ordinary category, i.e. all the structure morphisms are V-morphisms and the coherence conditions are fullfilled. The underlaying category of a symmetric monoidal V-category admits the structure of an ordinary symmetric monoidal 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 underlaying *category* [C,V] of V-functors admits a closed symmetric monoidal structure from the enriched one by taking the underlaying functor of each V-functor, the underlaying natural transformation of each V-natural transformation. Because a closed symmetric monoidal category is canonically enriched over 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 with 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 enrichments? Thank you in advance for any help. Tony