From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6624 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: question about monoidal categories Date: Tue, 19 Apr 2011 11:40:07 +1000 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1082) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1303217345 18251 80.91.229.12 (19 Apr 2011 12:49:05 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 19 Apr 2011 12:49:05 +0000 (UTC) Cc: categories@mta.ca To: claudio pisani Original-X-From: majordomo@mlist.mta.ca Tue Apr 19 14:49:01 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QCAMV-0007up-Jn for gsmc-categories@m.gmane.org; Tue, 19 Apr 2011 14:48:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35093) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QCAMD-00079f-LU; Tue, 19 Apr 2011 09:48:41 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QCAMA-00021S-Bu for categories-list@mlist.mta.ca; Tue, 19 Apr 2011 09:48:38 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6624 Archived-At: Dear Claudio, On 18/04/2011, at 8:37 PM, claudio pisani wrote: > Dear categorists, >=20 > suppose V is a monoidal category, with underlying category V_0, and X = is an ordinary category. Then (if I am not mistaken) the functor = category V_0^X has a monoidal structure, defined point-wise by that of = V and each functor f:X->Y gives a strong monoidal functor. >=20 > First question : supposing V closed, under which hypothesis is V_0^X = closed as well (as in the case V =3D Set)? >=20 This will be true if V is complete (as you suppose below anyway). = Writing, for simplicity, in the case where V is symmetric, the internal hom [f,g] for functors f,g:X->V_0 is given by the formula (which I hope = will be legible) [f,g]x =3D int_y [X(x,y).f(y),g(y)] Here X(x,y) is the hom-set in X, and X(x,y).f(y) is the coproduct of = X(x,y) copies of f(y), and int_y denotes the end over all object y in X. This is a special case of Brian Day's convolution structure where the = promonoidal structure on X is the ``cartesian'' one, corresponding to the cartesian closed structure on [X,Set]. > Second question: supposing that V_0^X is indeed closed and that V is = suitably complete, so that reindexing along f has a right adjoint = forall_f, then V_0^X is enriched over V by forall_X(A->B). How is this = enrichment related to the usual one of [X,v] when X is a V-category? >=20 They're the same, essentially because limits commute with limits.=20 All the best, Steve. > Best regards, >=20 > Claudio >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]