From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8309 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Is the category of group actions LCCC? Date: Thu, 4 Sep 2014 10:19:01 +1000 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Mac OS X Mail 7.3 \(1878.6\)) Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1409842745 30046 80.91.229.3 (4 Sep 2014 14:59:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 4 Sep 2014 14:59:05 +0000 (UTC) Cc: categories@mta.ca To: Timothy Revell Original-X-From: majordomo@mlist.mta.ca Thu Sep 04 16:58:59 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 1XPYV1-0005MF-Eo for gsmc-categories@m.gmane.org; Thu, 04 Sep 2014 16:58:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:41118) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XPYUG-0002pe-2r; Thu, 04 Sep 2014 11:58:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XPYUE-0003Of-Jl for categories-list@mlist.mta.ca; Thu, 04 Sep 2014 11:58:10 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8309 Archived-At: On 1 Sep 2014, at 7:12 pm, Timothy Revell = wrote: > I'm wondering whether the category of ALL group actions is locally > Cartesian closed.=20 This is what I answered Timothy: =3D=3D=3D=3D=3D=3D No, it=92s not. Since the category has a terminal object (1,1), being a LCCC would imply = it was cartesian closed. However, that would imply (G,X) \times =97 = preserved the initial object (1,0), which is false: (G,X)\times (1,0) =3D (G,0). =3D=3D=3D=3D=3D=3D But it seems there is more to the story.=20 The thing stopping the category of actions from being cartesian closed is that the category Gp of groups is not. = However, the category Gpd of groupoids and the category Cat of categories are. The (2-)category Cat//=92Set=92 of all category actions is defined as = follows: objects (A,F) are functors F : A =97> Set and morphisms (f,t) : (A,F) =97>= (B,G) are functors f : A =97> B with natural transformation t : F =3D=3D> G f. This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is ([B,C], K) where [B,C] is the functor category and K(g) =3D [B,Set](G, H = g). However Cat//=92Set=92 is not locally cartesian closed basically because = Cat is not. It is not even locally cartesian closed as a bicategory. The 2-category Gpd is cartesian closed; it is not locally cartesian = closed; it is locally cartesian closed as a bicategory.=20 Similarly, Gpd//=92Set=92 is locally cartesian closed as a bicategory. Often, in dealing with groups, we find groupoids help. This case is a good example and I hope helps in the applications you have in mind, Timothy. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]