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

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On Thu, Jun 21, 2012 at 6:34 AM, Michael Barr <barr@math.mcgill.ca> wrote:

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?

I don't know the original reference, but this is exercise 7H in The Joy of Cats, which contains a substantial hint for a proof (not involving a special case for order-2 elements, so maybe it is a different proof). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

text/plain;format=flowed;charset="iso-8859-1"; Content-Transfer-Encoding: 7bit Sender: categories@mta.ca Precedence: bulk Reply-To: "George Janelidze" <janelg@telkomsa.net> Dear Michael, My answer consists of several parts: 1. Indeed, Mac Lane gives no hints, and indeed, many people claim that Baer's result is unpublished. But look at Theorem 2 in [R. Baer, Absolute retracts in group theory, Bulletin AMS 52, 1946, 501-506] - it says: "The identity is the only group which is a retract of every containing group.", and its proof consists of one page. This fact, to which I refer below as not having non-trivial absolute retracts, of course immediately implies that there are no non-trivial injective objects. 2. Next, let us look at the last complete sentence on Page 21 and its explanation on the next page in [S. Eilenberg and J. C. Moore, Foundation of relative homological algebra, Memoirs AMS 55, 1965]. That sentence says "Not only is the category G [of groups] not injectively perfect, but we shall show that every injective object in G is trivial." There no reference to Bear's 19 year old (then) paper, but there is a reference on Baer's 31 year old paper (hence from 1934), where Baer constructs large simple groups. And of course knowing that there are arbitrary large simple groups with infinite cyclic subgroups makes the result trivial. 3. Next a surprise: look at Page 78 of [Maria S. Voloshina, On the holomorph of a discrete group, PhD Thesis, University of Rochester, 2003, arXiv:math/0302v2 [math.GR] 14 Jan 2004]. Surely not knowing about Baer's work, Voloshina says in the Abstract there: "In chapter 6 we give a short proof of the well-known fact due to S. Eilenberg and J. C. Moore that the only injective object in the category of groups is the trivial group." Her argument is something I have never seen before, and to me it is the best proof - so let me rephrase it here keeping the same notation but writing x* for the inverse of x: Let G be an injective group, x an element in G, F[a,b] and F[c,d] free groups on two-element sets {a,b} and {c,d} respectively, and i, f, g homomorphisms defined as follows: i : F[a,b] --> F[c,d] has i(a) = c and i(b) = dcd* f : F[a,b] --> G has f(a) = 1 and f(b) = x; g : F[c,d] --> G is any homomorphism with gi = f, which exists since G is injective. We have g(c) = gi(a) = f(a) = 1, and so x = f(b) = gi(b) = g(dcd*) = g(d)g(c)(g(d))* = g(d)(g(d))* = 1. Is not it wonderful, and is it possible that nobody have noticed it before 2003?! 4. Actually there is an important additional reason why I like Voloshina's argument. Let us look at her F[c,d] as Z+Z, that is, the coproduct of the additive group Z of integers with itself. Let p : Z+Z --> Z be the homomorphism induced by the identity homomorphism and the trivial homomorphism, and let k : ZbZ --> Z+Z be the kernel of p ("b" stands for "bemol"). This ZbZ occurs from a monad (Zb(-),e,m) on the category of groups, whose algebras are exactly Z-groups, and this is a special case of a categorical story presented in [D. Bourn and G. J., Protomodularity, descent, and semidirect products, TAC 4, 2, 1998, 37-46], with more categorical links in my papers with Francis Borceux and Max Kelly, and further developments by various authors. According to that story, k is the "the best example of a bad normal monomorphism" in a sense, and so it is a natural thing to use it "against injectivity". Back to Voloshina's notation, ZbZ is BETWEEN F[a,b] and F[c,d]. Indeed it is the subgroup in F[c,d] freely generated by the set S = {xcx* | x is an integer powers of d}, and I say "between" since S contains i(a) = c = 1c1* and i(b) = dcd*; that is, there is a homomorphism j : F[a,b] --> ZbZ with kj = i. Since ZbZ is free on S, Voloshina's homomorphism f extends to a homomorphism f' : ZbZ --> G, and so k : ZbZ --> Z+Z, which belongs to the categorical story, can be used instead of i in Voloshina's proof. 5. Another interesting thing is that k above is a normal monomorphism, and so my simple modification of Voloshina's argument proves that there are no non-trivial groups that are injective with respect to normal monomorphisms. And it is interesting because here we see the big difference between injective objects and absolute retracts in the category of groups. Indeed, going back to Baer's paper of 1946, we see "THEOREM 1. The group G is complete if, and only if, it meets the following requirement: (*) If G is a normal subgroup of the group E, then G is a retract of E." and then inside Remark on Page 503: "...This shows in particular that every group is a subgroup of a complete group..." Well, I must confess I have not checked Baer's proofs, but I hope they are correct. 6. Concerning your second question: What you saw (with a special argument for 2) is on the same Page 21 of the above-mentioned Eilenberg--Moore paper. The argument was used to prove that every epimorphism of groups is surjective with no mention of regular monomorphisms, but in fact they prove that, for every homomorphism j : H --> G, there exist two homomorphisms k, l : G --> P with k(g) = L(g) only when g is in j(H). This also appears as Exercise 5 of Section 5 of Chapter I in Mac Lane's "Categories for the Working Mathematician", with very precise hints. However, a group theorist would probably prefer to use known facts pushouts of group monomorphisms. Regards, George P.S. While I was writing this, several other answers came. But instead of changing my message, let me only mention that it is good to have more references, and that Maria Nogin should be the same person as Maria Voloshina (Probably one of the two surnames is her husband's surname). -------------------------------------------------- From: "Michael Barr" <barr@math.mcgill.ca> Sent: Thursday, June 21, 2012 3:34 PM To: "Categories list" <categories@mta.ca> Cc: "Bob Raphael" <raphael@alcor.concordia.ca> Subject: categories: Two questions

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

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