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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?).

Though I am assuming $G$ torsors are defined over a site. But do they mean that the notion of $G$ principal bundles and $G$ torsors over a site are indistinguishable in their set up?

Or

Is there any separate meaning for $G$ Principal bundles for them?

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 over $B$ will correspond to an element of $H^1(B,G)$.

Thank you.

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    $\begingroup$ Did you try reading beyond the first paragraph of the first section? $\endgroup$
    – Mike Shulman
    Commented May 15, 2020 at 1:10
  • $\begingroup$ @MikeShulman Sir, I came across the notion of Principal 2 bundles from Bartels 2 bundle's (arxiv.org/abs/math/0410328) where a principal 2 bundle is defined within a category internal to some generalised smooth space and the structure group is a coherent 2-group as introduced in arxiv.org/abs/math/0307200. Baez, Schrieber in arxiv.org/abs/hep-th/0412325 introduced a notion of connection in it but restricted the base to dengenerate smooth 2 space and Loop spaces. Wockel in arxiv.org/abs/0803.3692 also mentioned about Principal 2 bundles. $\endgroup$
    – Adittya Chaudhuri
    Commented May 15, 2020 at 12:52
  • $\begingroup$ @MikeShulman Though Wockel mentioned about the general Principal 2 bundles but his treatment was mostly on what they called semi strict principal 2 bundle over a manifold(considered as a degenerate smooth 2 space). Though I don't have much background in infinity category but in the paper arxiv.org/abs/0803.3692 from the 4th paragraph in the 1st section it seems that Principal infinity bundles is a generalisation of Principal 2 bundles when the structure group is something like infinity space(which they called geometric infinity groups). $\endgroup$
    – Adittya Chaudhuri
    Commented May 15, 2020 at 12:58
  • $\begingroup$ @MikeShulman Now since I don't have have much background in infinity category I am not able to understand what geometric infinity group is!! I am also confused about the term geometric group . But it seems geometric 2 groups must be related to 2 groups (in the sense of arxiv.org/abs/math/0307200). Also since in the previous papers I mentioned about principal 2- bundles above they clear mentioned about their base (i.e some 2 space) but here I could not get what is the base of such Principal infinity bundles!! Is it some $(\infty,1)$ category? $\endgroup$
    – Adittya Chaudhuri
    Commented May 15, 2020 at 13:04
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    $\begingroup$ I don't know what the point being made in of all your comments is; I was just saying that I think your original question is answered a few pages later in the paper. $\endgroup$
    – Mike Shulman
    Commented May 15, 2020 at 23:38

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