A short note on coproducts of abelian pro-Lie groups (Q6582384)

In this paper, the authors introduce the notion of conditional coproduct of pro-Lie groups and show that the Cartesian product of abelian pro-Lie groups can be characterized by its universal property.\N\NFor a convergent family of pro(jective)-Lie groups \(\{\tau_j: A_j\to G\}_{j\in J }\) the \textit{conditional coproduct} is a topological group \(G\) that admits the universal property that for every other convergent family \(\{\psi_j: A_j\to H\}_{j\in J }\), there is a unique morphism \(\omega: G \to H\) such that \(\psi_j = \omega\circ \tau_j, \forall j\in J\) (Definition 3). The main result is Theorem 4 stating that 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 \(\tau_j : A_j \to P\).