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Cocycle categories
 In Algebraic Topology
, 2009
"... Suppose that G is a sheaf of groups on a space X and that Uα ⊂ X is an open covering. Then a cocycle for the covering is traditionally defined to be a family of elements gαβ ∈ G(Uα ∩ Uβ) such that gαα = e and gαβgβγ = gαγ when all elements are restricted to the group G(Uα ∩ Uβ ∩ Uγ). ..."
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Suppose that G is a sheaf of groups on a space X and that Uα ⊂ X is an open covering. Then a cocycle for the covering is traditionally defined to be a family of elements gαβ ∈ G(Uα ∩ Uβ) such that gαα = e and gαβgβγ = gαγ when all elements are restricted to the group G(Uα ∩ Uβ ∩ Uγ).
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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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) →
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
Pro objects in simplicial presheaves
, 2009
"... This paper describes model structures for the category of pro objects in simplicial presheaves on an arbitrary small Grothendieck site, all based on the injective model structure for simplicial presheaves. The most fundamental of these structures is an analogue of the Edwards ..."
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This paper describes model structures for the category of pro objects in simplicial presheaves on an arbitrary small Grothendieck site, all based on the injective model structure for simplicial presheaves. The most fundamental of these structures is an analogue of the Edwards
Model structures for prosimplicial presheaves
, 2011
"... This paper describes various model structures for the category of proobjects in simplicial presheaves on an arbitrary small Grothendieck site. The most fundamental of these structures is a generalization of the EdwardsHastings model structure for prosimplicial sets [5], in which a cofibration is ..."
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This paper describes various model structures for the category of proobjects in simplicial presheaves on an arbitrary small Grothendieck site. The most fundamental of these structures is a generalization of the EdwardsHastings model structure for prosimplicial sets [5], in which a cofibration is a
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.