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Pointed torsors
, 2010
"... 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) → ..."
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Cited by 5 (4 self)
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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) →
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... 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 shif ..."
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Cited by 3 (3 self)
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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 ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles 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 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
Galois descent criteria
, 2010
"... Suppose that X is a presheaf of Kan complexes on the étale site etk 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 e ..."
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Cited by 2 (2 self)
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Suppose that X is a presheaf of Kan complexes on the étale site etk 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.
Cosimplicial spaces and cocycles
, 2010
"... 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 ..."
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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
The Verdier hypercovering theorem
, 2010
"... 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. ..."
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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.
The homotopy classification of gerbes
, 2006
"... 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 2groupoids consisting of all sheaves of groups, their isomorphisms and homotopies. ..."
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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 2groupoids 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 nonabelian cohomology object H 2 (C, L) of equivalence classes of gerbes with band L.
Lecture 08 (January 6, 2011)
"... 18 Torsors for groups Suppose that G is a sheaf of groups. A Gtorsor is traditionally defined to be a sheaf X with a free Gaction such that X/G ∼ = ∗ in the sheaf category. The requirement that the action G × X → X is free means that the isotropy subgroups of G for the action are trivial in all ..."
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18 Torsors for groups Suppose that G is a sheaf of groups. A Gtorsor is traditionally defined to be a sheaf X with a free Gaction such that X/G ∼ = ∗ in the sheaf category. The requirement that the action G × X → X is free means that the isotropy subgroups of G for the action are trivial in all sections, which is equivalent to requiring that all sheaves of fundamental groups for the Borel construction EG ×G X are trivial. There is an isomorphism of sheaves ˜π0(EG ×G X) ∼ = X/G. Also the simplicial sheaf EG ×G X is the nerve of a sheaf of groupoids, which is given in each section by the translation category for the action of G(U) on X(U); this means, in particular, that all sheaves of higher homotopy groups for EG ×G X vanish. It follows that a Gsheaf X is a Gtorsor if and only if the map EG ×G X → ∗ is a local weak equivalence. 1 Example 18.1. The Borel construction EH ×H H = EH for a group H is the nerve of the translation category for the action H × H → H which is given by the multiplication of H. There is a unique map e h − → h for all h ∈ H, so that EH ×H H is a contractible simplicial set. If G is a sheaf of groups, then EG ×G G is contractible in each section, so that the map EG ×G G → ∗ is a local weak equivalence, and G is a Gtorsor. This object is often called the trivial Gtorsor. Example 18.2. Suppose that L/k is a finite Galois extension with Galois group G. Then the étale covering Sp(L) → Sp(k) has Čech resolution C(L) and there is an isomorphism of simplicial schemes C(L) ∼ = EG ×G Sp(L). The simplicial presheaf map
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"... Suppose that C is a small Grothendieck site. As before, write s Pre(C) for the category of simplicial presheaves on the site C. The discussion that follows will be confined to simplicial presheaves. It has an exact analog for simplicial sheaves. Let A be a fixed choice of simplicial presheaf. The sl ..."
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Suppose that C is a small Grothendieck site. As before, write s Pre(C) for the category of simplicial presheaves on the site C. The discussion that follows will be confined to simplicial presheaves. It has an exact analog for simplicial sheaves. Let A be a fixed choice of simplicial presheaf. The slice category A/s Pre(C) has all morphisms x: A → X as objects, and all diagrams as morphisms. X