Results 1 
5 of
5
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). ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
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).
Homotopy theories of diagrams
, 2011
"... The work which is displayed in this paper arose from a preliminary study of the homotopy theory of dynamical systems. In general, a dynamical system consists of an action X × S → X ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The work which is displayed in this paper arose from a preliminary study of the homotopy theory of dynamical systems. In general, a dynamical system consists of an action X × S → X
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. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
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