From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6628 Path: news.gmane.org!not-for-mail From: claudio pisani Newsgroups: gmane.science.mathematics.categories Subject: Re: question about monoidal categories Date: Wed, 20 Apr 2011 10:43:13 +0100 (BST) Message-ID: Reply-To: claudio pisani 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: dough.gmane.org 1303300858 7953 80.91.229.12 (20 Apr 2011 12:00:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 20 Apr 2011 12:00:58 +0000 (UTC) Cc: categories@mta.ca To: Steve Lack Original-X-From: majordomo@mlist.mta.ca Wed Apr 20 14:00:53 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 1QCW5U-0003DT-Q5 for gsmc-categories@m.gmane.org; Wed, 20 Apr 2011 14:00:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49463) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QCW5P-0001Sz-A1; Wed, 20 Apr 2011 09:00:47 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QCW5L-0001Mn-OX for categories-list@mlist.mta.ca; Wed, 20 Apr 2011 09:00:44 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6628 Archived-At: Dear Steve, your answer has been very useful to me. Thus, for a closed complete category V, there are two related doctrines: 1) the V-categories [X,V], indexed by V-Cat; 2) the V-categories V_0^X, indexed by Cat.=20 The former is "included" in the latter via the forgetful functor=20 V-Cat -> Cat, and the latter has more structure since it is also an indexed= monoidal category. I would like to know if the second doctrine, and its relationships with the= first one, had been considered explicitly somewhere (perhaps in some paper= by Brian Day?). Thank you again Claudio =A0=20 --- Mar 19/4/11, Steve Lack ha scritto: > Da: Steve Lack > Oggetto: Re: categories: question about monoidal categories > A: "claudio pisani" > Cc: categories@mta.ca > Data: Marted=EC 19 Aprile 2011, 03:40 > Dear Claudio, >=20 > On 18/04/2011, at 8:37 PM, claudio pisani wrote: >=20 >> 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=A0 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 >=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) >=20 > [f,g]x =3D int_y [X(x,y).f(y),g(y)] >=20 > 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. >=20 > 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]. >=20 >> Second question: supposing that V_0^X is=A0 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]=A0 when X is a V-category? >>=20 >=20 > They're the same, essentially because limits commute with > limits.=20 >=20 > All the best, >=20 > Steve. >=20 >=20 >> Best regards, >>=20 >> Claudio >>=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]