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Higher principal bundles. (English) Zbl 1109.18007

If \(\mathcal{C}\) is a small Grothendieck site and \(G\) a presheaf of simplicial groups, then a \(G\)-torsor is a cofibrant simplicial presheaf \(X\) such that the canonical map \(X/G\) is a local trivial fibration. The authors also extend this definition to presheaves of simplicial groupoids with discrete objects.
The objective of the paper is to give a description of the corresponding homotopy theoretic invariant \([*,\overline{W}G] \cong [*,dBG] \), where \(\overline{W}G\) and \(dBG\) are models for a classifying space construction for \(G\). \(\overline{W}G\) is the universal cocycles construction which is a generalization of the classical construction of Eilenberg-Mac Lane, and \(dBG\) is the diagonal of the bisimplicial object arising from the standard nerve functor applied to \(G\).
It is proved in the paper that, if \(G\) is a groupoid enriched in simplicial sets, then \(\overline{W}G\) and \(dBG\) are weakly equivalent.
First, the authors give the following classification result concerning torsors for presheaves of simplicial groups: If \(G\) is a presheaf of simplicial groups, then there exists a natural bijection \([*,\overline{W}G] \cong \pi_0(G\text{-}{{\mathcal T}\!ors}_0)\), where \(G\text{-}{{\mathcal T}\!ors}_0\) is the category of cofibrant simplicial \(G\)-presheaves \(X\) such that the map \(X/G \to *\) is a local weak equivalence.
After that they prove the main result of the paper: If \(G\) is a presheaf of simplicial groupoids with discrete objects, then there is a natural bijection \([*,dBG] \cong \pi_0(G\text{-}{{\mathcal T}\!ors}_0)\), where \(\pi_0(G\text{-}{{\mathcal T}\!ors}_0)\) is the set of path components of the category of \(G\)-torsors \(G\text{-}{{\mathcal T}\!ors}_0\).
The paper finishes with an application of these results. They identify the class of path components of the category of \(G\)-gerbes for a sheaf of groups \(G\) with the set of isomorphism classes of right torsors over the automorphism 2-groupoid object. This theorem can be inferred from a result of L. Breen [The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 401–476 (1990; Zbl 0743.14034)].

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
14F35 Homotopy theory and fundamental groups in algebraic geometry

Citations:

Zbl 0743.14034
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