  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.
