From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7342 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: Two questions Date: Thu, 21 Jun 2012 18:07:27 +0100 Message-ID: References: Reply-To: Ronnie Brown NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1340319570 12179 80.91.229.3 (21 Jun 2012 22:59:30 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 21 Jun 2012 22:59:30 +0000 (UTC) Cc: Categories list , Bob Raphael To: Michael Barr Original-X-From: majordomo@mlist.mta.ca Fri Jun 22 00:59:29 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ShqLX-0003sl-S1 for gsmc-categories@m.gmane.org; Fri, 22 Jun 2012 00:59:27 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50099) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1ShqKf-0001OW-Ir; Thu, 21 Jun 2012 19:58:33 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ShqKh-0001QO-DF for categories-list@mlist.mta.ca; Thu, 21 Jun 2012 19:58:35 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7342 Archived-At: With regard to the second question, the problem seems to be for epic. I found the solution somewhere and put it as exercise 8 of section 6.1 of my Topology and Groupoids book (all editions). Here is the outline argument (I have not checked it lately!): Prove that in the category $\grp$ of groups, a morphism $f : G \to H$ is monic if and only if it is injective; less trivially, $f$ is epic if and only if $f$ is surjective. [Suppose $f$ is not surjective and let $K = \Im f$. If the set of cosets $H/K$ has two elements, then $K$ is normal in $H$ and it is easy to prove $f$ is not epic. Otherwise there is a permutation $\gamma$ of $H/K$ whose only fixed point is $K$. Let $\pi : H \to H/K$ be the projection and choose a function $\theta : H/K \to H$ such that $\pi \theta = 1$. Let $\tau : H \to K$ be such that $x = (\tau x)(\theta\pi x)$ for all $x$ in $H$ and define $\lambda : H \to H$ by $x \mapsto (\tau x) (\theta\gamma\pi x)$. The morphisms $\alpha, \beta$ of $H$ into the group $P$ of all permutations of $H$, defined by $\alpha(h)(x) = hx$, $\beta(h) = \lambda^{-1}\alpha(h)\lambda$ satisfy $\alpha h = \beta h$ if and only if $h \in K$. Hence $\alpha f = \beta f$]. Ronnie On 21/06/2012 14:34, Michael Barr wrote: > Googling around, I have come on several claims that there are no > non-trivial injectives in the category of groups (e.g., Mac Lane in the > 1950 Duality for groups paper credits Baer with an elegant proof, but > gives no hint of what it might be and Baer's earlier paper on injectives > doesn't mention it). I have not come on any proof of this, however. > > Somewhere I have seen a proof that all monics in the category of groups > are regular. I think it was in a paper by Eilenberg and ??? and it > needed > a special argument if there were elements of order 2. Can someone > help me > find this? > > Michael > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]