From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8311 Path: news.gmane.org!not-for-mail From: pjf Newsgroups: gmane.science.mathematics.categories Subject: Re: Is the category of group actions LCCC Date: Thu, 04 Sep 2014 09:16:51 -0400 Message-ID: References: Reply-To: pjf NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1409842935 32436 80.91.229.3 (4 Sep 2014 15:02:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 4 Sep 2014 15:02:15 +0000 (UTC) Cc: categories@mta.ca To: Timothy Revell Original-X-From: majordomo@mlist.mta.ca Thu Sep 04 17:02:09 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 1XPYY4-0007n3-Me for gsmc-categories@m.gmane.org; Thu, 04 Sep 2014 17:02:08 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:41130) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XPYXb-00036t-63; Thu, 04 Sep 2014 12:01:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XPYXZ-0003SP-Qe for categories-list@mlist.mta.ca; Thu, 04 Sep 2014 12:01:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8311 Archived-At: On page -15 (yes, a negative page number) of the TAC reprinting of Abelian Categories (in the 2003 Forward) I wrote: The very large category [described] in Exercise 6-A -- with a few variations -- has been a great source of counterexamples over the years....In its category of abelian-group objects Ext(A,B) is a proper class iff there???s a non-zero group homomorphism from A to B (it needn???t respect the actions) hence the only injective object is the zero object (which settled a once-open problem about whether there are enough injectives in the category of abelian groups in every elementary topos with natural-numbers object. http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf On 2014-09-01 05:12, Timothy Revell wrote: > Dear All, > > 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. > > - The objects are pairs (G,X), where G is a group and X is a G-Set. > > - 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 > > h(g) * f(x) = f(g * x) > > where * on the left denotes the group action of G' on X' and * on the > right denotes the group action of G on X. > > > All the best, > Tim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]