nLab
Whitehead principle of nonabelian cohomology

Idea

What is called nonabelian cohomology is the general intrinsic cohohomology of any (∞,1)-topos $H$ with coefficients in any object $A\in H$, 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 $A\in H$ has a decomposition as a sequence of objects

$$A\to \cdots \to A_3\to A_2\to A_1\to A_0\to *,$$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 $n$-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 $BG$ in that we have fiber sequences

$$B^2{K}_1\to A_2\to BG=A_1$$\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 $X\to BG$ and an abelian $B^{2}K$-cocycle on its total space

(…)

Examples

A string structure is a nonabelian cocycle with coefficients in the string 2-group. This is equivalently a $BU(1)$-cocycle (a bundle gerbe) on the total space of the underlying $\mathrm{Spin}$-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, Stacks and non-abelian cohomology (pdf)