From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6683 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Enriched adjoint functor theorem? Date: Mon, 23 May 2011 21:56:17 +0200 Message-ID: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (iPad Mail 8H7) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1306220607 11028 80.91.229.12 (24 May 2011 07:03:27 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 24 May 2011 07:03:27 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue May 24 09:03:23 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 1QOleF-0003HN-6W for gsmc-categories@m.gmane.org; Tue, 24 May 2011 09:03:23 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51037) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QOlbA-0002YC-Jt; Tue, 24 May 2011 04:00:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QOlb5-0004Dz-Oy for categories-list@mlist.mta.ca; Tue, 24 May 2011 04:00:08 -0300 Original-Content-Transfer-Encoding: quoted-printable Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6683 Archived-At: By Yoneda, what you are asking for is an isomorphism x@(a@b) =3D~ (x@a)@b since V(x,[a@b,c]) =3D~ V(x@(a@b), c) and V(x,[a,[b,c]]) =3D~ V((x@a)@b,c). In general, I see no way around proving a certain associativity constraint i= nvertible. Monoidal categories give promonoidal categories via p(a,b;c) =3D V(a@b,c). On the other hand, closed categories almost give promonoidal categories via p= (a,b;c) =3D V(a,[b,c]) except that the associativity constraint may not be i= nvertible. Ross Begin forwarded message: > Subject: Re: categories: Re: Enriched adjoint functor theorem? >=20 >> ------ Original Message ------ >> Received: Mon, 23 May 2011 02:59:46 AM EDT >> From:=20 >> To: categories@mta.ca >> Subject: categories: Enriched adjoint functor theorem? >>>=20 >>>=20 >>> I have a closed category V with internal hom-functpr [-,-], and I am >>> trying to show that it is *monoidal* closed. I was able to prove (using >>> the adjoint functor theorem) that the hom-functor [a,-] has a left-adjoi= nt >>> L^a: V --> V, but in order to obtain a monoidal closed strucutre, one >>> needs to have a natural isomorphism in V: >>>=20 >>> [L^a(b),c] -=3D- [b,[a,c]] (*) >>>=20 >>> This will also imply associativity and coherence. >>>=20 >>> So, I am asking if there is a way to prove (*) based on some form of >>> enriched adjoint functor theorem, without figuring out the structure >>> of L^a(b) explicitly. >>>=20 >>>=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]