On the dearth of coproducts in the category of locally compact groups

TL;DR

This paper investigates the existence of coproducts in the category of locally compact groups, proving their scarcity when involving connected groups, and provides auxiliary results on characteristic indices of such groups.

Contribution

It establishes the non-existence of coproducts for certain families of locally compact groups and introduces new properties of characteristic indices in Lie groups.

Findings

Coproducts are rare in the category of locally compact groups involving connected groups.

Characteristic indices decrease when passing to semisimple closed Lie subgroups.

Characteristic indices also decrease along dense-image morphisms.

Abstract

We prove that a family of at least two non-trivial, almost-connected locally compact groups cannot have a coproduct in the category of locally compact groups if at least one of the groups is connected; this confirms the intuition that coproducts in said category are rather hard to come by, save for the usual ones in the category of discrete groups. Along the way we also prove a number of auxiliary results on characteristic indices of locally compact or Lie groups as defined by Iwasawa: that characteristic indices can only decrease when passing to semisimple closed Lie subgroups, and also along dense-image morphisms.