Abstract
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.
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In [3] the second author and 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 of profinite groups exists and agrees with the cartesian product. The latter, namely \(\prod _{i\in I}A_i\), for any family \((A_i)_{i\in I}\) of profinite abelian groups, as has shown Neukirch in [5], has the 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 a family of abelian pro-Lie groups. For formulating our result, we need to adapt the concepts, originally introduced for classes of profinite groups by Neukirch in [5] (see also [6, D.3]) to the category of abelian pro-Lie groups. We shall use additive notation, in particular, 0 denotes the identity element. Our notations and the terminology are standard in topological group theory and follow [2, 6].
Let \((A_j)_{\in J}\) be a family of topological groups, H a topological group, and \({\mathbb {F}}\) a family of continuous homomorphisms \(\phi _j:A_j\rightarrow H\). We say that \({\mathbb {F}}\) is convergent, if for every neighborhood U of the identity of H the set \(J_U:= \{j\in J: \phi _j(A_j)\not \le U\}\) is finite.
FormalPara Example 2For any family \((A_j)_{j\in J}\) of topological groups let \(H=\prod _{j\in J}A_j\) be the cartesian product with the product topology. Then the family \({\mathbb F}=(\tau _j)_{j\in J}\) of canonical embeddings \(\tau _j:A_j\rightarrow H\), given by \(\tau _j(a)=(a_l)_{l\in J}\) where \(a_l=a\) if \(l=j\) and else \(a_l=0\), is convergent. This is immediate from the fact that given an open subgroup N of P, then by the very definition of the cartesian product, almost all factors \(\tau _j(A_j)\) must be subgroups of N.
We define the conditional product by means of a universal property, resembling the one of the coproduct (direct sum) of abelian discrete groups:
For a convergent family \(\tau _j:A_j\rightarrow G\), \(j\in J\) of morphisms in a category \({{\mathcal {A}}}\) of topological groups we call G a conditional coproduct of \((A_j)_{j\in J}\) if for every convergent family of morphisms \(\psi _j:A_j\rightarrow H\) in \({\mathcal {A}}\) there is a unique morphism \(\omega :G\rightarrow H\) such that \(\psi _j=\omega \circ \tau _j\) for all \(j\in J\).
We have the following Theorem:
In the category of abelian pro-Lie groups, the conditional coproduct of a family \((A_j)_{j\in J}\) of abelian pro-Lie groups is the cartesian product \(P:=\prod _{j\in J} A_j\) together with the canonical embeddings \(\tau _j:A_j\rightarrow P\).
The next result is well-known, see e.g. [4, 1. Theorem (g), p. 68]:
Let \(f:X\rightarrow Y\) be a continuous map and \(A\subseteq X\). Then
Conditional coproducts are unique in the following sense:
If G and \(G'\) are conditional coproducts of a family \((A_j)_{j\in J}\) of topological groups in a category \({\mathcal {A}}\) for the convergent families \(\tau _j:A_j\rightarrow G\) and \(\tau '_j:A_j\rightarrow G'\), \(j\in J\) of morphisms in \({\mathcal {A}}\), then there is an isomorphism \(\lambda :G\rightarrow G'\) such that \(\tau '_j=\lambda \circ \tau _j\) for all \(j\in J\).
FormalPara ProofLetting in Definition 3\(G'\) play the role of H and the \(\tau _j':A_j\rightarrow G'\) play the role of the \(\psi _j\) we may conclude the existence of \(\omega :P\rightarrow P'\) such that
Likewise, letting G play the role of H and \(\tau _j\) the role of \(\psi _j\) we obtain
Equations (1) and (2) together imply that
Set
It is an immediate consequence of Eq. (3) that
We apply Lemma 5 to \(\omega :G\rightarrow G'\) and \(A=Q\) and, noting that \(\omega (Q)=Q'\), obtain the equality
Next consider the family of morphisms \(\chi _j:A_j\rightarrow P'/\overline{Q'}\), for \(j\in J\), where \(\chi _j\) sends \(A_j\) to \(\{0\}\). Certainly the family \((\chi _j)_{j\in J}\) is convergent. Hence the universal property of \(P'\) renders a unique morphism \(\omega '':G'\rightarrow G'/\overline{Q'}\) such that
The unique candidate for \(\omega ''\) must therefore map all of P to \(\{0\}\). Taking for \(\omega ''\) the quotient map from \(P'\) onto \(G'/\overline{Q'}\) also satifies the commutativity relations and hence as claimed, \(\omega :G\rightarrow G'\) is surjective. Therefore we have from the definitions of Q and \(Q'\) in Eqs. (4) and (6) that
Now Eq. (5) shows that
so that indeed \(\omega ':G'\rightarrow G\) is an isomorphism.
We conclude the proof by remarking that \(\lambda \) can be taken to be \(\omega \). \(\square \)
We note that for profinite groups the conditional coproduct agrees with the free pro-\({\mathcal {C}}\) product for \({\mathcal {C}}\) the variety of abelian profinite groups, see [5, 6].
We recall that for the cartesian product \(P=\prod _{j\in J}A_j\) of a family of topological groups \(A_j\) there is a unique family \(\tau _j:A_j\rightarrow P\) of continuous homomorphisms defined as \(\tau _j(a_j)=(x_i)_{i\in J}\) where \(x_i=1\) for \(i\ne j\) and \(x_j=a_j\). These homomorphisms are sometimes called canonical embeddings.
Let \({\mathcal {A}}\) be the category of topological abelian pro-Lie groups (i.e. groups which are projective limits of Lie groups: see [1, pp. 160ff. and Chapter 5]). Each pro-Lie group G has a filter basis \({{\mathcal {N}}}(G)\) of closed normal subgroups such that G/N is a Lie group, and \(G\cong \lim _{N\in {{\mathcal {N}}}(G)}G/N\). (See e.g. [1, p.160, Definition A.])
Recall that every locally compact abelian group is a pro-Lie group, every almost connected locally compact group is a pro-Lie group. Trivially, then, every profinite group is a pro-Lie group. Every cartesian product \(P=\prod _{j\in J}A_j\) of pro-Lie groups \(A_j\) is itself a pro-Lie group.
Let H be a pro-Lie group and \({\mathbb F}\) be a convergent family of morphisms \(\psi _j:A_j\rightarrow H\). Then, for each \(N\in {{\mathcal {N}}}(H)\), the set \(\{j\in J:\psi _j(A_j)\not \subseteq N\}\) is finite.
FormalPara ProofLet \(N\in {{\mathcal {N}}}(H)\). The Lie group H/N has an identity neighborhood V in which \(\{0\}=N/N\) is the only subgroup of H/N. Now let \(p:H\rightarrow H/N\) be the quotient morphism and set \(U=p^{-1}(V)\).
Therefore \(\psi _j(A_j)\not \subseteq N\) implies \(\psi _j(A_j)\not \subseteq U\). However the set of j satisfying this condition is finite by Definition 1 applied to the conditional coproduct of the family \((A_j)_{j\in J}\). This completes the proof of the Lemma. \(\square \)
FormalPara Proof of Theorem 4The uniqueness, up to isomorphism, of the conditional coproduct follows from Proposition 1.
Thus, according to Definition 3, we need to show that given \(H\in {\mathcal {A}}\) and a convergent family of morphisms \(\psi _j:A_j\rightarrow H\) then there exists a unique morphism \(\omega :P\rightarrow H\) with \(\psi _j=\omega \circ \tau _j\) for all \(j\in J\).
We note first that every \(x\in P\) has a presentation
for unique elements \(a_j\in A_j\). Denote by \({\mathcal {N}}(H)\) the set of all closed subgroups of H such that H/N is a Lie group. It is a consequence of [1, Theorem 3.27] that \({\mathcal {N}}(H)\) is a filter basis of closed subgroups of H and that
algebraically and topologically.
Fix \(N\in {\mathcal {N}}(H)\) and let let \(J_N:=\{j\in J:\psi _j(A_j)\not \le N\}\). Then, by Lemma 6, the set \(J_N\) is finite and, taking the presentation Eq. (7) for \(x\in P\) and \(\tau _j(A_j)\le N\) for all \(j\notin J_N\) into account, we obtain a well-defined morphism \(\omega _N:P\rightarrow H/N\) by letting
For subgroups \(M\le N\) of H, both in \({\mathcal {N}}(H)\), let \(\pi _{MN}:H/M\rightarrow H/N\) denote the canonical epimorphism.
For \(M\le N\) 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 \(H=\varprojlim _{N\in {\mathcal {U}}}H/N\) implies the existence of a unique continuous homomorphism \(\omega :P\rightarrow H\), which satisfies the desired relations
\(\square \)
1 Final comments
A coproduct of a family of objects in a category \({\mathcal {A}}\) 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 \({\mathcal {A}}\) 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 \(\mathbb {Z}(2)\) and \(\mathbb {Z}(3)\) is PSL\((2,\mathbb {Z})\).
In any category \({\mathcal {A}}\) with a well-introduced dual category, such as the category of locally compact abelian groups, the coproduct \(\coprod _j A_j\) of a family \(A_j\), \(j\in J\), is naturally isomorphic to the dual \({{\widehat{P}}}\) of \(P:=\prod _j \widehat{A_j}\), 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 \({\mathcal {A}}\) is the category of profinite abelian groups, then its dual is the category \({\mathcal {T}}\) of abelian torsion groups. The product in \({\mathcal {T}}\) of a family of torsion groups \(T_j\) is the torsion group \(\textrm{tor}(\prod _j T_j)\) of the cartesian product. So by the time we arrive at the coproduct of, say, an unbounded family of cyclic groups \(A_j\) in \({\mathcal {A}}\), we may have a complicated object \(\coprod _{j\in J} A_j\) 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 [1]). 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.
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Communicated by John S. Wilson.
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Herfort, W., Hofmann, K.H. & Russo, F.G. A short note on coproducts of Abelian pro-Lie groups. Monatsh Math (2023). https://doi.org/10.1007/s00605-023-01915-1
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DOI: https://doi.org/10.1007/s00605-023-01915-1