On epimorphisms in some categories of infinite-dimensional Lie groups

TL;DR

This paper investigates the nature of epimorphisms in categories of infinite-dimensional Lie groups, showing that certain inclusions are epimorphisms in some categories but not in others, revealing nuanced structural differences.

Contribution

It demonstrates that the inclusion of stabilizer subgroups in diffeomorphism groups are epimorphisms in the category of smooth Lie groups modeled on complete locally convex spaces, contrasting with finite-dimensional cases.

Findings

Inclusion of stabilizer groups in Diff X is an epimorphism in the category of smooth Lie groups.

Epimorphisms between finite-dimensional Lie groups have dense range.

In Banach-Lie groups, proper closed subgroup inclusions are not epimorphisms.

Abstract

Let $X$ be a smooth compact connected manifold. Let $G=\mbox{Diff}\, X$ be the group of diffeomorphisms of $X$, equipped with the $C^\infty$-topology, and let $H$ be the stabilizer of some point in $X$. Then the inclusion $H\to G$, which is a morphism of two regular Fr\'echet--Lie groups, is an epimorphism in the category of smooth Lie groups modelled on complete locally convex spaces. At the same time, in the latter category, epimorphisms between finite dimensional Lie groups have dense range. We also prove that if $G$ is a Banach--Lie group and $H$ is a proper closed subgroup, the inclusion $H\to G$ is not an epimorphism in the category of Hausdorff groups.