Principal infinity-bundles -- General theory

Abstract

The theory of principal bundles makes sense in any infinity-topos, such as that of topological, of smooth, or of otherwise geometric infinity-groupoids/infinity-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure infinity-group G these G-principal infinity-bundles reproduce the theories of ordinary principal bundles, of bundle gerbes/principal 2-bundles and of bundle 2-gerbes and generalize these to their further higher and equivariant analogs. The induced associated infinity-bundles subsume the notions of gerbes and higher gerbes in the literature. We discuss here this general theory of principal infinity-bundles, intimately related to the axioms of Giraud, Toen-Vezzosi, Rezk and Lurie that characterize infinity-toposes. We show a natural equivalence between principal infinity-bundles and intrinsic nonabelian cocycles, implying the classification of principal infinity-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber infinity-bundles associated to principal infinity-bundles subsumes a theory of infinity-gerbes and of twisted infinity-bundles, with twists deriving from local coefficient infinity-bundles, which we define, relate to extensions of principal infinity-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice infinity-topos. In a companion article [NSSb] we discuss explicit presentations of this theory in categories of simplicial (pre)sheaves by hyper-Cech cohomology and by simplicial weakly-principal bundles; and in [NSSc] we discuss various examples and applications of the theory.

49 pages; v2: published version; v3: missing assumption added to Lemma 3.9

References in corpus (4)

• Higher Topos Theory
• Principal infinity-bundles - Presentations
• Segal topoi and stacks over Segal categories
• On the classification of fibrations

Cited by in corpus (53)

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• Twisted Cohomotopy implies M5-brane anomaly cancellation
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• Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5
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• Global Double Field Theory is Higher Kaluza-Klein Theory
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• Principal $\infty$-Bundles and Smooth String Group Models
• Higher Structures in M-Theory
• Higher geometry for non-geometric T-duals
• Higher Gauge Theory
• Twisted smooth Deligne cohomology
• Cyclification of Orbifolds
• Principal actions of stacky Lie groupoids
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• Atiyah-Segal completion for the Hermitian K-theory of Symplectic Groups
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• Diffeological Principal Bundles and Principal Infinity Bundles