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"Gerbe" is a construct in homological algebra and topology. They can be seen as a generalization of principal bundles to the setting of 2-categories. "Gerbe" is a French (and archaic English) word that literally means wheat sheaf. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2.

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Second nonabelian group cohomology: cocycles vs. gerbes

The general abstract story of nonabelian cocycle, infinity-gerbes and associated infinity-bundles is in section 4 of Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal infinity-bundles - General …
Urs Schreiber's user avatar