A collection of three articles on bundles in higher geometry:
Thomas Nikolaus, Urs Schreiber, Danny Stevenson,
Principal $\infty$-bundles
( talk handout (6 pages, 8 pages) )
General Theory,
to appear in Journal of Homotopy and Related Structures, 2014 ((DOI) 10.1007/s40062-014-0083-6, arXiv:1207.0248)
Presentations,
Journal of Homotopy and Related Structures, March 2014 ((DOI) 10.1007/s40062-014-0077-4, arXiv:1207.0249)
Applications (not out yet, but see section 5 here)
This is a sub-project in the context of differential cohomology in a cohesive topos.
Abstract The theory of $G$-principal bundles makes sense in any (∞,1)-topos, such as that of topological or of smooth ∞-groupoids and more general in any slices of these. It provides a geometric model for structured structrued higher nonabelian (sheaf hyper-) cohomology and controls general fiber bundles in terms of associated bundles. For suitable group objects $G$ these $G$-principal ∞-bundles reproduce the theory of ordinary principal bundles , of principal 2-bundles , of gerbes and 2-gerbes , of bundle gerbes and bundle 2-gerbes and generalizes them to higher analogs of arbitrary degree. The induced associated ∞-bundles subsume the notion of Giraud's gerbes, Breen's 2-gerbes, Lurie's $n$-gerbes, and generalize these to the notion of nonabelian ∞-gerbes; which are the universal local coefficient bundles for nonabelian twisted cohomology.
We discuss the general abstract theory of principal ∞-bundles, observe that it is induced directly by the ∞-Giraud axioms that characterize(∞,1)-toposes. A central result is a natural equivalence between principal ∞-bundles and intrinsic nonabelian cocycles, implying the classification of principal $\infty$-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber ∞-bundles associated to principal $\infty$-bundles subsumes a theory of ∞-gerbes and of twisted ∞-bundles, with twists deriving from local coefficient ∞-bundles, which we define, relate to extension of principal $\infty$-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice (∞,1)-topos.
Abstract We discuss two aspects of the presentation of the theory of principal ∞-bundles in an (∞,1)-topos in terms of categories of simplicial (pre)sheaves.
First we show that over a ∞-cohesive site $C$ and for $G$ a presheaf of simplicial groups which is $C$-acyclic, $G$-principal ∞-bundles over any object in the (∞,1)-topos over $C$ are classified by hyper-Cech cohomology with coefficients in $G$. Then we show that over a site $C$ with enough points, principal ∞-bundles in the (∞,1)-topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete ∞-groupoids) and the smooth site (in smooth ∞-groupoids, generalizing Lie groupoids and differentiable stacks).
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differential cohomology in a cohesive topos
Principal ∞-bundles – models and general theory