Infinite dimensional manifolds from a new point of view

Abstract

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of categories and functors. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the sense that de Rham cohomology and singular cohomology can be naturally defined and the basic properties (Functorial Property, Homotopy Invariant, Mayer–Vietoris Sequence) are preserved. In this setting we define the classifying space BG of a Lie group G as an infinite dimensional manifold. Using simplicial homotopy theory and the Chern–Weil theory for principal G-bundles we show that de Rhamʼs theorem holds for BG when G is compact. Finally we get, as an unexpected byproduct, two simplicial set models for the classifying spaces of compact Lie groups; they are totally different from the classical models constructed by Milnor, Milgram, Segal and Steenrod.