nLab Whitehead principle of nonabelian cohomology

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)-toposes.

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 sequences

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

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

• Bertrand Toën, p. 8 of: Stacks and Non-abelian cohomology, lecture at Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory, MSRI 2002 (slides, ps, pdf, pdf)