Direct limit Lie groups and manifolds

Abstract

We show that every countable strict directed system of finitedimensional Lie groups has a direct limit in the category of smooth Lie groups modelled on sequentially complete, locally convex spaces. Similar results are obtained for countable directed systems of finite-dimensional manifolds, and for countable directed systems of finite-dimensional Lie groups and manifolds over totally disconnected local fields. An uncountable strict directed system of finite-dimensional Lie groups has a direct limit in the category of Lie groups in the sense of convenient differential calculus, provided certain technical hypotheses are satisfied.