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A manifold has
a set of orientations;
an xyz of topological spin structures
a 3-groupoid of topological string structures;
a 7-groupoid of topological fivebrane stuctures, etc.
and for some reason it is common in the literature (which of course is small in the last cases) to speak of these $n$-groupoids, but not so common to speak of the xyz here:
Namely the homotopy fiber of the second Stiefel-Whitney class
$Spin(X) \to Top(X,B SO) \stackrel{(w_2)_*}{\to} Top(X, B^2 \mathbb{Z}_2) \,.$I have added one reference that explicitly discusses the groupoid of spin structures to spin structure.
Do you have further references?
@Urs This looks worth pursuing. It may relate to HQFTs. We can discuss this when you arrive (in Lisbon).
How is it going in Lisbon? I hope it’s sunny. I need some sun. Though today it seems winter is taking a break even here.
Urs, you skipped Lisboa ?
Urs, you skipped Lisboa ?
No, I’ll go. My flight is day after tomorrow, Feb 9.
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