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

Let G be a Lie group which is the union of an ascending sequence G1 ⊆ G2 ⊆··· of Lie groups (all of which may be infinite-dimensional).We study the question when G = lim −→Gn in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C∞-diffeomorphisms of a σ-compact smooth manifoldM; and for test function groups C∞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. © 2007 Elsevier Inc. All rights reserved.