Vladimir Uspenskiy

On epimorphisms in some categories of topological groups

A morphism f\colon X\to Y in a category {\mathbf K} is an epimorphism if for every object Z in {\mathbf K} the induced map f^*:\mathrm{Mor}(Y,Z)\to \mathrm{Mor}(X,Z) is injective. It is known that epimorphisms in the category of Hausdorff groups need not have a dense range. We investigate the question whether epimorphisms in certain categories of infinite-dimensional Lie groups must have a dense range. For example, let G be the group of diffeomorphisms of a compact smooth manifold X, equipped with the C^\infty-topology, and let H\subset G be the stabilizer of a point p\in X. Then the inclusion H\to G is an epimorphism in the category of Lie groups modeled on Fr\’echet spaces but not an epimorphism in the category of Hausdorff groups. If H is a proper closed subgroup of a Banach-Lie group G, then the inclusion H\to G is not an epimorphism in the category of Hausdorff groups, but may be an epimorphism in the category of Banach-Lie groups. This is a joint work with Vladimir Pestov, arXiv:2102.07276