Infinite dimensional manifolds from a new point of view

TL;DR

This paper introduces a novel approach to infinite dimensional manifolds using category theory, extending classical concepts like cohomology and classifying spaces, and provides new models for classifying spaces of compact Lie groups.

Contribution

It offers a new categorical framework for infinite dimensional manifolds, generalizes classical cohomology theories, and constructs novel simplicial models for classifying spaces.

Findings

De Rham and singular cohomology are naturally defined for the new manifolds.

The paper proves de Rham's theorem for the classifying space BG.

Two new simplicial set models for classifying spaces of compact Lie groups are constructed.

Abstract

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. 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 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 Rham's theorem holds for $BG$. Finally we get, as an unexpected byproduct, two new simplicial set models for the classifying spaces of compact Lie groups; it is totally different from the classical models constructed by Milnor Milgram, Segal and Steenrod.