A spherical Milnor space construction yields projective quotients and double quotients that encode Z2-twisted principal bundles and obstruction classes in low-degree cohomology.
Breen,On the Classification of 2-Gerbes and 2-Stacks, Ast´ erisque, vol
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Spherical Milnor Spaces II: Projective Quotients and Higher Topological Structures
A spherical Milnor space construction yields projective quotients and double quotients that encode Z2-twisted principal bundles and obstruction classes in low-degree cohomology.