nLab
Whitehead principle of nonabelian cohomology

Contents

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

Last revised on January 27, 2017 at 02:07:06. See the history of this page for a list of all contributions to it.