From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6684 Path: news.gmane.org!not-for-mail From: Gabor Lukacs Newsgroups: gmane.science.mathematics.categories Subject: Re: Enriched adjoint functor theorem? Date: Mon, 23 May 2011 15:23:35 -0500 (CDT) Message-ID: Reply-To: Gabor Lukacs NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII; FORMAT=flowed X-Trace: dough.gmane.org 1306220628 11176 80.91.229.12 (24 May 2011 07:03:48 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 24 May 2011 07:03:48 +0000 (UTC) Cc: categories@mta.ca To: Ross Street Original-X-From: majordomo@mlist.mta.ca Tue May 24 09:03:44 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QOlea-0003W6-1h for gsmc-categories@m.gmane.org; Tue, 24 May 2011 09:03:44 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51042) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QOlc1-0002bT-Sa; Tue, 24 May 2011 04:01:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QOlby-0004FC-QA for categories-list@mlist.mta.ca; Tue, 24 May 2011 04:01:02 -0300 Content-ID: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6684 Archived-At: Hi Ross, On Mon, 23 May 2011, Ross Street wrote: > By Yoneda, what you are asking for is an isomorphism > > x@(a@b) =~ (x@a)@b You are quite right. The reason that I prefer to seek [a,[b,c]] =~ [a@b,c] is because Kelly showed in "Tensor Products in Categories" that the latter implies associativity, as well as coherence (Theorem 11). While I understand the structure of [-,-] very well (and for example, I was able to show that [a,[b,c]] =~ [b,[a,c]]), I do not know much about the structure of a@b, and I expect its structure to be extremely complicated. > In general, I see no way around proving a certain associativity > constraint invertible. My question could be rephased as: Is there an extra condition on [-,-], which can be expressed only using [-,-], that will imply associativity of @ ? Of course, the alternative way of phrasing this was to seek a left-V-adjoint for [a,-]. That was the starting point for my question. Best, Gabi [For admin and other information see: http://www.mta.ca/~cat-dist/ ]