Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Abstract

Let G be a Lie group which is the union of an ascending sequence of Lie groups G_n (all of which may be infinite-dimensional). We study the question when G is the direct limit of the G_n's in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for G the group Diff_c(M) of compactly supported smooth diffeomorphisms of a sigma-compact smooth manifold M, and for test function groups C^infty_c(M,H) of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.