Principal infinity-bundles - Presentations

TL;DR

This paper explores presentations of principal infinity-bundles within an infinity-topos, providing classification methods via hyper-Cech-cohomology and explicit simplicial bundle constructions, with applications to smooth and discrete sites.

Contribution

It introduces new classification and presentation techniques for principal infinity-bundles in an infinity-topos, extending to smooth and discrete sites.

Findings

Principal infinity-bundles classified by hyper-Cech-cohomology.

Bundles presented by stalkwise weak equivalences in sheaf topos.

Explicit descriptions for discrete and smooth sites.

Abstract

We discuss two aspects of the presentation of the theory of principal infinity-bundles in an infinity-topos, introduced in [NSSa], 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 infinity-bundles over any object in the infinity-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 infinity-bundles in the infinity-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 infinity-groupoids) and the smooth site (in smooth infinity-groupoids, generalizing Lie groupoids and differentiable stacks). In the companion article [NSSc] we use these presentations for constructing classes of examples of (twisted) principal infinity-bundles and for the discussion of various applications.