nLab
Whitehead principle of nonabelian cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

What is called nonabelian cohomology is the general intrinsic cohohomology of any (∞,1)-topos H\mathbf{H} with coefficients in any object AHA \in \mathbf{H}, not necessarily an Eilenberg-MacLane object.

But there is a general notion of Postnikov tower in an (∞,1)-category that applies in any locally presentable (∞,1)-category, in particular in (∞,1)-toposes.

This implies that every object AHA\in \mathbf{H} has a decomposition as a sequence of objects

AA 3A 2A 1A 0*, A \to \cdots \to A_3 \to A_2\to A_1 \to A_0 \to * \,,

where A kA_k is an kk-truncated object, in fact the nn-truncation of AA.

This implies that every nn-truncated connected object AA is given by a possibly nonabelian 0-truncated group object GG and a sequence of abelian extensions of the delooping BG\mathbf{B}G in that we have fiber sequences

B 2K 1A 2BG=A 1 \mathbf{B}^2 K_1 \to A_2 \to \mathbf{B}G = A_1

etc.

(…)

It follows that any cocycle XA 2X \to A_2 decomposes into the principal bundle classified by XBGX \to \mathbf{B}G and an abelian B 2K\mathbf{B}^2 K-cocycle on its total space

(…)

Examples

A string structure is a nonabelian cocycle with coefficients in the string 2-group. This is equivalently a BU(1)\mathbf{B}U(1)-cocycle (a bundle gerbe) on the total space of the underlying SpinSpin-principal bundle. See the section In terms of classes on the total space.

References

The term “Whitehead principle” for nonabelian cohomology is used in

Created on November 9, 2010 00:07:51 by Urs Schreiber (87.212.203.135)