From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8308 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Is the category of group actions LCCC? Date: Wed, 3 Sep 2014 11:01:36 +1000 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Mac OS X Mail 7.3 \(1878.6\)) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1409748181 25039 80.91.229.3 (3 Sep 2014 12:43:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 3 Sep 2014 12:43:01 +0000 (UTC) Cc: categories@mta.ca To: Timothy Revell Original-X-From: majordomo@mlist.mta.ca Wed Sep 03 14:42:54 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XP9tk-0007lC-Qf for gsmc-categories@m.gmane.org; Wed, 03 Sep 2014 14:42:52 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39676) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XP9t6-0002FT-69; Wed, 03 Sep 2014 09:42:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XP9t4-00053m-M5 for categories-list@mlist.mta.ca; Wed, 03 Sep 2014 09:42:10 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8308 Archived-At: Dear Tim, Write Act for the category you mention. Then Act/(G,0) is clearly = equivalent to Grp/G. Since Grp/G is not cartesian closed, neither is Act/(G,0). Regards, Steve Lack. On 1 Sep 2014, at 7:12 pm, Timothy Revell = wrote: > Dear All, >=20 > I'm wondering whether the category of ALL group actions is locally > Cartesian closed. This is NOT the functor category [G,Set] for some > category G with one object, since we allow G to vary. To be more > specific the category is as follows. >=20 > - The objects are pairs (G,X), where G is a group and X is a G-Set. >=20 > - A morphism (G,X) -> (G', X') is given by a pair (h,f), where = h:G->G' > is a group homomorphism and f: X -> X' is a function (a morphism in = Set) > such that for all g in G, x in X >=20 > h(g) * f(x) =3D f(g * x) >=20 > where * on the left denotes the group action of G' on X' and * on the > right denotes the group action of G on X. >=20 >=20 > All the best, > Tim >=20 >=20 > --=20 > Timothy Revell, > Department of Computer and Information Sciences, > University of Strathclyde. > The University of Strathclyde is a charitable body, registered in > Scotland, with registration number SC015263. >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]