A short note on coproducts of abelian pro-Lie groups

The notion of conditional coproduct of a family of abelian pro-Lie groups in the category of abelian pro-Lie groups is introduced. It is shown that the cartesian product of an arbitrary family of abelian pro-Lie groups can be characterized by the universal property of the conditional coproduct.

In [HM77] the second author and S. Morris have provided criteria when the (co)product of a family of locally compact abelian groups exists.For profinite abelian groups the product of any family (A i ) i∈I of profinite groups exists and agrees with the cartesian product P := i∈I A i .J. Neukirch has shown in [Neu71] that P has a universal property resembling that of a coproduct (direct sum) in the category of (discrete) abelian groups.In the present note we present a version of his result, valid for cartesian products of the much larger family of abelian pro-Lie groups (see [HM07,Ch. 5]).For formulating our result, we need to adapt the concepts, originally introduced for families of profinite groups by J. Neukirch in [Neu71] (see also [RZ10,D.3]), to the category of abelian pro-Lie groups.
Definition 1.Let (A j ) ∈J be a family of topological groups, H a topological group, and F = (φ j ) j∈J , φ j : A j → H, a family of continuous homomorphisms.We say that F is convergent, if for every identity neighborhood U of H the set J U := {j ∈ J : φ j (A j ) ⊆ U} is finite.
Example 2. For any family (A j ) j∈J of topological groups let H = j∈J A j be the cartesian product with the Tychonov topology.Let the family F = (τ j ) j∈J of natural morphisms τ j : A j → H be given by τ j (a) = (a k ) k∈J with a j = a and a k = 0 otherwise.Then F is convergent.This follows immediately from the definition of the product topology on H.The morphisms τ j are called the natural embeddings.
We define the conditional coproduct by means of a universal property, resembling the one of the coproduct (direct sum) of abelian discrete groups: Definition 3. In a category A of topological groups we call G a conditional coproduct of the family (A j ) j∈J of objects if there is a convergent family τ j : A j → G, j ∈ J of morphisms such that for every convergent family of morphisms ψ j : A j → H, j ∈ J in A there is a unique morphism ω : G → H such that ψ j = ω • τ j for all j ∈ J.The morphisms τ j are called the coprojections of the conditional coproduct.
We shall prove the following Theorem: Theorem 4. In the category of abelian pro-Lie groups, the conditional coproduct of a family (A j ) j∈J of abelian pro-Lie groups is the cartesian product P := j∈J A j for the canonical embeddings τ j : A j → P .
But first we secure the uniqueness of the conditional coproduct: Proposition 1.If G and G ′ are conditional coproducts of a family (A j ) j∈J of topological groups in a category A for the convergent families τ j : A j → G and τ ′ j : A j → G ′ , j ∈ J of morphisms in A, then there is a natural isomorphism λ : G → G ′ such that τ ′ j = λ • τ j for all j ∈ J. Proof.By Definition 2, since G is a conditional coproduct of the family (A j ) j∈J with the coprojections τ j , j ∈ J, there is a unique morphism Therefore, by (1) and (2), we have However, trivially we also have, Therefore, by the uniqueness in Definition 3, from (3) and (4) we have (5) Now by exchanging the roles of G and G ′ we also have Hence by ( 5) and (6), λ is an isomorphism, which we had to show.
We note that for profinite groups the conditional coproduct agrees with the free pro-C product for C the variety of abelian profinite groups, see [Neu71,RZ10].
Let A be the category of topological abelian pro-Lie groups (i.e.groups which are projective limits of Lie groups: see [HM07,pp.Recall that every locally compact abelian group is a pro-Lie group, every almost connected locally compact group is a pro-Lie group by Yamabe's Theorem.Trivially, then, every profinite group is a pro-Lie group.Every cartesian product P = j∈J A j of pro-Lie groups A j is itself a pro-Lie group. Lemma 5. Let H be an abelian pro-Lie group and F be a convergent family of morphisms ψ j : A j → H.Then, for each N ∈ N (H), the set {j ∈ J : ψ j (A j ) ⊆ N} is finite.
Proof.Let N ∈ N (H).The Lie group H/N has an identity neighborhood V in which {0} is the only subgroup of H/N.Now let p : H → H/N be the quotient morphism and set U = p −1 (V ).
Therefore ψ j (A j ) ⊆ N implies ψ j (A j ) ⊆ U. However the set of j satisfying this condition is finite by Definition 1 applying to the conditional coproduct of the family (A j ) j∈J .This completes the proof of the Lemma.
Proof of Theorem 4. The uniqueness, up to isomorphism, of the conditional coproduct follows from Proposition 1.
Thus, according to Definition 3, we need to show that given an abelian pro-Lie group H and a convergent family of morphisms ψ j : A j → H then there exists a unique morphism ω : P → H with ψ j = ω • τ j for all j ∈ J.
We note first that every x ∈ P has a presentation (7) x = (τ j (a j )) j∈J for unique elements a j ∈ A j .Fix N ∈ N (H) and let let J N := {j ∈ J : ψ j (A j ) ≤ N}.Then, by Lemma 5, the set J N is finite and, taking the presentation Eq. ( 7) for x ∈ P and τ j (A j ) ≤ N for all j / ∈ J N into account, we obtain a well-defined morphism ω N : P → H/N by letting For subgroups M ≤ N of H, both in N (H), let π N M : H/M → H/N denote the canonical epimorphism.
For M ≤ N one obtains from Eq. ( 9) the compatibility relation as depicted in the following diagram: Taking the relations in Eq. ( 10) into account we see that the universal property of the inverse limit H = lim ← −N∈U H/N implies the existence of a unique continuous homomorphism ω : P → H, which satisfies the desired relations (11) (∀j ∈ J) ψ j = ω • τ j .

Notes.
A coproduct of a family of objects in a category A is a product in the category obtained by reversing all arrows.Curiously, while products are usually considered simple concepts, coproducts are often tricky in many categories A other than the category of abelian groups.Therefore, in conclusion of this note, a few general comments may be in order.
One of the early surprises is that in the familiar category of groups, the coproduct of Z(2) and Z(3) is PSL(2, Z).
In any category A with a well-introduced dual category, such as the category of locally compact abelian groups, the coproduct j A j of a family A j , j ∈ J, is naturally isomorphic to the dual P of P := j A j , the product of its duals.
Even in special cases, such as the case of compact abelian groups A j , the result is a complicated coproduct, since the character group of an infinite product of discrete abelian groups may be hard to deal with.
If A is the category of profinite abelian groups, then its dual is the category T of abelian torsion groups.The product in T of a family of torsion groups T j is the torsion group tor( j T j ) of the cartesian product.So by the time we arrive at the coproduct of, say, an unbounded family of cyclic groups A j in A, we may have a complicated object j∈J A j in our hands.Therefore, any special situation may be welcome, where a coproduct is lucid-even when its scope of application may be restricted.An example of such a situation is our present conditional coproduct in the rather large yet reasonably well-understood category of abelian pro-Lie groups (see Chapter 5 of [HM07]).The authors encountered such a coproduct in a study of certain locally compact abelian p-groups.Our conditional coproduct covers a somewhat restricted supply of families of morphisms which we call "convergent".Here we encounter the rather extraordinary event that for each of such families their conditional coproduct agrees with their cartesian product.Classically, one is familiar with a situation of coproducts agreeing with products in the category of finite abelian groups which, after all, is rather special.
160ff.and Chapter 5]).Each pro-Lie group G has a filterbasis N (G) of closed normal subgroups such that G/N is a Lie group, and G ∼ = lim N ∈N (G) G/N.(See e.g.[HM07, p.160, Definition A.])