From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8313 Path: news.gmane.org!not-for-mail From: Clemens.BERGER@unice.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: Is the category of group actions LCCC? Date: Thu, 04 Sep 2014 18:00:21 +0200 Message-ID: References: Reply-To: Clemens.BERGER@unice.fr 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 1409877948 13242 80.91.229.3 (5 Sep 2014 00:45:48 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 5 Sep 2014 00:45:48 +0000 (UTC) Cc: Timothy Revell , categories@mta.ca To: Ross Street Original-X-From: majordomo@mlist.mta.ca Fri Sep 05 02:45:43 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 1XPhem-0000rk-8w for gsmc-categories@m.gmane.org; Fri, 05 Sep 2014 02:45:40 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42395) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XPhdt-0000JK-92; Thu, 04 Sep 2014 21:44:45 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XPhdt-0006Zb-9T for categories-list@mlist.mta.ca; Thu, 04 Sep 2014 21:44:45 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8313 Archived-At: Hi Timothy, hi Ross, this message to highlight the importance of the construction of the category of group actions (Grp,C) over an arbirary base category C (while Timothy is just considering the case C=Sets). Indeed, the category of group actions (Grp,C) has as objects triples (G,X,\phi) consisting of a group G, an object X of C, and a monoid morphism G->C(X,X), and as maps (g,f):(G,X)->(H,Y) pairs consisting of a group morphism g:G->H and an arrow f:X->Y in C such that an obvious pentagonal diagram in Sets commutes. Composition in this category is defined by gluing together two pentagons along a commutative square. This construction seems to have quite interesting preservation properties (alas not inclucing cartesian nor local cartesian closedness). These properties may have their use in the theory of semi-abelian categories. Just a few samples: The category (Grp,Grp) of group actions in groups is equivalent to Bourn's category of split epimorphisms in Grp, also known as the category of points in Grp. The latter is known to inherit several important properties of the category Grp, such as protomodularity, regularity and exactness. In general, if C is regular (exact) then (Grp,C) as well. In the theory of cocommutative Hopf algebras over a field k, a structure thm of Gabriel-Cartier may be interpreted as saying that over an algebraically closed field of characteristic 0, the category CocommHopf_k is equivalent to (Grp,Lie_k), where Lie_k is the category of Lie k-algebras. Since the latter is exact, this shows that CocommHopf_k is exact as well. An adjunction argument implies then that CocommHopf_k is exact for any field of characteristic 0. This is the most difficult step in showing that CocommHopf_k is actually semi-abelian for any field of characterstic zero. All the best, Clemens. Le 2014-09-04 02:19, Ross Street a ??crit??: > 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. > > This is what I answered Timothy: > ====== > No, it???s 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 ??? > preserved > the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0). > ====== > > But it seems there is more to the story. > 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//???Set??? of all category actions is defined as > follows: > objects (A,F) are functors F : A ???> Set and morphisms (f,t) : (A,F) ???> > (B,G) > are functors f : A ???> B with natural transformation t : F ==> 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) = [B,Set](G, H > g). > > However Cat//???Set??? 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. > > Similarly, Gpd//???Set??? 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/ ]