Schreiber Principal ∞-bundles -- theory, presentations and applications

A pair of two articles on bundles in higher geometry:

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• Thomas Nikolaus, Urs Schreiber, Danny Stevenson:

  Principal $\infty$-bundles

  ( talk handout (6 pages, 8 pages) )

  1. General Theory,

     Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801

     download:

     • doi:10.1007/s40062-014-0083-6

     • arXiv:1207.0248

  2. Presentations,

     Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622

     download:

     • doi:10.1007/s40062-014-0077-4

     • arXiv:1207.0249

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The first article works axiomatically in the style of “synthetic homotopy theory” without need to specify any set-based model for (∞,1)-categories, while the second article shows how to model/present everything in suitable simplicial model categories or fibration categories.

A formalization of various of the synthetic arguments of the first article in homotopy type theory was later given in:

• Ulrik Buchholtz, Floris van Doorn, Egbert Rijke: Higher Groups in Homotopy Type Theory [arXiv:1802.04315]

Further discussion and generalization to equivariant bundles:

• Hisham Sati, Urs Schreiber, Equivariant principal ∞-bundles [arXiv:2112.13654]

1) General theory

Abstract. The theory of $G$-principal bundles makes sense in any (∞,1)-topos, such as that of topological or of smooth ∞-groupoids and, more generally, in any slices of these. It provides a geometric model for structured 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, and 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.

2) Presentations

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).

Errata

• Lemma 3.9 in “General theory” (p. 22) is wrong as stated. It holds true under unstated the assumption that there is an essentially unique morphism $\ast \to \mathbf{B}G$. This assumption typically holds in gros toposes where the terminal object does look like a point, but crucially not in typical petit toposes such as that of $\infty$-sheaves over a non-contractible topological space or scheme. This unstated assumption crept in with the last line of Remark 2.22 (p. 15).

  Hence to fix the lemma one should make explicit the statement of this assumption, whence the lemma becomes more of a remark. No further statements in the article are affected.

  We thank Eivind Otto Hjelle for catching this.