Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Classifying Space of a Topological 2-Group

John C. Baez, Danny Stevenson

TL;DR

The paper extends classical bundle theory to principal 2-bundles by developing nonabelian Čech cohomology with coefficients in a topological 2-group \\mathcal{G} and proving a classification theorem: under mild hypotheses, \\check{H}^1(M,\\mathcal{G}) \\cong [M|\\mathcal{G}|], where |\\mathcal{G}| is the geometric realization of the nerve. It consolidates multiple viewpoints on topological 2-groups, provides an elementary proof of the classifying-bundle correspondence, and applies the framework to the string 2-group to extract rational characteristic classes via a transgression from H^*(BG). The results unify and extend prior work of Jur\\c{o}, Bartels, and BBK, with implications for higher gauge theory and higher-gerbe classifications.

Abstract

Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and Cech cohomology with coefficients in a topological 2-group C, also known as 'nonabelian cohomology'. Then we give an elementary proof that under mild conditions on M and C there is a bijection between the first Cech cohomology of M with coefficients in C and [M,B|C|] where B|C| is the classifying space of the geometric realization of the nerve of C. Applying this result to the 'string 2-group' String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of the rational cohomology of BG modulo the ideal generated by c, where c is any nonzero element in the 4th cohomology of BG.

The Classifying Space of a Topological 2-Group

TL;DR

The paper extends classical bundle theory to principal 2-bundles by developing nonabelian Čech cohomology with coefficients in a topological 2-group \\mathcal{G} and proving a classification theorem: under mild hypotheses, \\check{H}^1(M,\\mathcal{G}) \\cong [M|\\mathcal{G}|], where |\\mathcal{G}| is the geometric realization of the nerve. It consolidates multiple viewpoints on topological 2-groups, provides an elementary proof of the classifying-bundle correspondence, and applies the framework to the string 2-group to extract rational characteristic classes via a transgression from H^*(BG). The results unify and extend prior work of Jur\\c{o}, Bartels, and BBK, with implications for higher gauge theory and higher-gerbe classifications.

Abstract

Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and Cech cohomology with coefficients in a topological 2-group C, also known as 'nonabelian cohomology'. Then we give an elementary proof that under mild conditions on M and C there is a bijection between the first Cech cohomology of M with coefficients in C and [M,B|C|] where B|C| is the classifying space of the geometric realization of the nerve of C. Applying this result to the 'string 2-group' String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of the rational cohomology of BG modulo the ideal generated by c, where c is any nonzero element in the 4th cohomology of BG.

Paper Structure

This paper contains 10 sections, 8 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Topological $2$-Groups
  4. Nonabelian Cohomology
  5. Proofs
  6. Proof of Theorem \ref{['classifying']}
  7. Remarks on Theorem \ref{['classifying']}
  8. Proof of Lemma \ref{['lem1']}
  9. Proof of Lemma \ref{['lem2']}
  10. Proof of Lemma \ref{['lem3']}

Key Result

Theorem 0

Let $G$ be a topological group. Then there is a topological space $BG$ with the property that for any paracompact Hausdorff space $M$, there is a bijection

Theorems & Definitions (9)

  • Theorem 0
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Lemma 3