The first paragraph of the section **Overview** in the paper *Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson* https://arxiv.org/abs/1207.0248 precisely reads the following:

*The concept of a G-principal bundle for a topological or Lie group G is fundamental in classical topology and differential geometry,. More generally, for G a geometric group in the sense of a sheaf of groups over some site, the notion of G- principal bundle or G-torsor is fundamental in topos theory. Its relevance rests in the fact that G-principal bundles constitute natural geometric representatives of cocycles in degree 1 nonabelian cohomology H1(−,G) and that general fiber bundles are associated to principal bundles.*

**My confusion is the following:**

 Since they are working with a sheaf of groups $G$ over a site(which they call here geometric group) so it is natural to ask what did they mean by $G$ Principal  bundle or $G$-torsor in this context(Note neither they have  mentioned about the "base space" of the $G$ Principal bundle nor about     where $G$-torsors are defined(like over a topological space or a site?)). Also they said $G$ Principal  bundles  constitute natural geometric representatives of cocycles in degree 1 nonabelian cohomology H1(−,G) . Note instead of writing $H^1(X,G)$ they wrote $H^1(-,G)$. So my guess is if $B$ is the base space of the $G$ Principal bundle then an isomorphism class of such $G$ principal  bundle will correspond to an element of $H^1(B,G)$.

 


Thank you.