This paper gives a characterization of homotopy fibres over trivial torsors of the inverse image maps π ∗ : B(H − Tors) → B(π ∗ H − tors) of torsor categories which are induced by geometric morphisms π: Shv(C) →

Suppose that X is a presheaf of Kan complexes on the étale site et|k of a field k. Let j: X → GX be an injective fibrant model for X: this means that j is a local weak equivalence and that GX has the extension property with respect to all monomorphisms of simplicial presheaves which are local weak equivalences.

Gerbes are locally connected presheaves of groupoids on a small Grothendieck site C. Gerbes are classified up to local weak equivalence by path components of a cocycle category taking values in the diagram Grp(C) of 2-groupoids consisting of all sheaves of groups, their isomorphisms and homotopies. If F is a full subpresheaf of Grp(C) then the set [∗, BF] of morphisms in the homotopy category of simplicial presheaves classifies gerbes locally equivalent to objects of F up to weak equivalence. If St(πF) is the stack completion of the fundamental groupoid πF of F, if L is a global section of St(πF), and if FL is the homotopy fibre over L of the canonical map BF → B St(πF), then [∗, FL] is in bijective correspondence with Giraud’s non-abelian cohomology object H 2 (C, L) of equivalence classes of gerbes with band L.

We introduce nonabelian differential cohomology classifying ∞-bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian n-group B n−1 U(1). Notable examples are String 2-bundles [9] and Fivebrane 6-bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spin-structures to String-structures [13] and further to Fivebrane-structures [133, 52], are abelian Chern-Simons 3- and 7-bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞-Lieintegrating the L∞-algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2- and twisted Fivebrane 6-bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted K-theory. We explain the Green-Schwarz mechanism in heterotic string theory in terms of twisted String 2-bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6-bundles. We close by transgressing differential cocycles to mapping

The Verdier hypercovering theorem is a traditional and widely used method of approximating the morphisms [X, Y] between two objects in homotopy categories of simplicial sheaves and presheaves by simplicial homotopy classes of maps.

This paper is a retelling of the basic homotopy theory of cosimplicial spaces, from a point of view that is informed by sheaf theoretic homotopy theory. The overall plan is to interpolate ideas associated with the injective model structure for cosimplicial spaces with classical results of Bousfield and Kan. The