Principal ∞-bundles: presentations

We discuss two aspects of the presentation of the theory of principal -bundles in an -topos, introduced in Nikolaus et al. (Principal -bundles: general theory, 2012), in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site and for a presheaf of simplicial groups which is -acyclic, -principal -bundles over any object in the -topos over are classified by hyper-ech-cohomology with coefficients in . Then we show that over a site with enough points, principal -bundles in the -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). In the companion article (Nikolaus et al. in Principal -bundles: examples and applications, 2012) we use these presentations for constructing classes of examples of (twisted) principal -bundles and for the discussion of various applications.