nLab
Whitehead principle of nonabelian cohomology
Contents
Context
Cohomology
cohomology

Special and general types
Special notions
Variants
Operations
Theorems
$(\infty,1)$ -Topos Theory
(∞,1)-topos theory

elementary (∞,1)-topos

(∞,1)-site

reflective sub-(∞,1)-category

(∞,1)-category of (∞,1)-sheaves

(∞,1)-topos

(n,1)-topos , n-topos

(∞,1)-quasitopos

(∞,2)-topos

(∞,n)-topos

hypercomplete (∞,1)-topos

over-(∞,1)-topos

n-localic (∞,1)-topos

locally n-connected (n,1)-topos

structured (∞,1)-topos

locally ∞-connected (∞,1)-topos , ∞-connected (∞,1)-topos

local (∞,1)-topos

cohesive (∞,1)-topos

structures in a cohesive (∞,1)-topos

Contents
Idea
What is called nonabelian cohomology is the general intrinsic cohohomology of any (∞,1)-topos $\mathbf{H}$ with coefficients in any object $A \in \mathbf{H}$ , 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)-topos es.

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

$A \to \cdots \to A_3 \to A_2\to A_1 \to A_0 \to *
\,,$

where $A_k$ is an $k$ -truncated object, in fact the $k$ -truncation of $A$ .

This implies that every $n$ -truncated connected object $A$ is given by a possibly nonabelian 0-truncated group object $G$ and a sequence of abelian extensions of the delooping $\mathbf{B}G$ in that we have fiber sequence s

$\mathbf{B}^2 K_1 \to A_2 \to \mathbf{B}G = A_1$

etc.

(…)

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

(…)

Examples
A string structure is a nonabelian cocycle with coefficients in the string 2-group . This is equivalently a $\mathbf{B}U(1)$ -cocycle (a bundle gerbe ) on the total space of the underlying $Spin$ -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 September 10, 2020 at 07:20:56.
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