Re: Two questions

["George Janelidze" <janelg@telkomsa.net>]

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Baer's result is unpublished.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
>
> --
> The United States has the best congress money can buy.



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]