A short note on coproducts of abelian pro-Lie groups
type: Item
instance of: scholarly article
title: A short note on coproducts of abelian pro-Lie groups
author: Wolfgang Herfort
author: Karl H. Hofmann
author: Francesco Russo
published in: Monatshefte für Mathematik
publication date: 2024-08-02T00:00:00Z
review text: 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 \(\tau_j : A_j \to P\).
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 22E20
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 7891500
zbMATH Keywords: pro-Lie groups
zbMATH Keywords: coproducts, conditional coproduct of pro-Lie groups
MaRDI profile type: Publication
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DOI: 10.1007/S00605-023-01915-1
description: scientific article; zbMATH DE number 7891500