Whitehead principle of nonabelian cohomology
Special and general types
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
What is called nonabelian cohomology is the general intrinsic cohohomology of any (∞,1)-topos with coefficients in any object , 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 has a decomposition as a sequence of objects
where is an -truncated object, in fact the -truncation of .
This implies that every -truncated connected object is given by a possibly nonabelian 0-truncated group object and a sequence of abelian extensions of the delooping in that we have fiber sequences
It follows that any cocycle decomposes into the principal bundle classified by and an abelian -cocycle on its total space
A string structure is a nonabelian cocycle with coefficients in the string 2-group. This is equivalently a -cocycle (a bundle gerbe) on the total space of the underlying -principal bundle. See the section In terms of classes on the total space.
The term “Whitehead principle” for nonabelian cohomology is used in
Revised on January 27, 2017 02:07:06
by Urs Schreiber