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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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Is the first differential Pontryagin class a morphism of stacks?
simplifies, and it can be shown that in this case it actually gives a morphism of stacks from the classifying stack of principal $G$-bundles with connections to the (higher) stack of principal $U(1)$-$n$-gerbes … with connections to $U(1)$-2-gerbes with connections)? …