## Cocycle categories (2009)

Venue: | In Algebraic Topology |

Citations: | 10 - 6 self |

### BibTeX

@INPROCEEDINGS{Jardine09cocyclecategories,

author = {J. F. Jardine},

title = {Cocycle categories},

booktitle = {In Algebraic Topology},

year = {2009},

pages = {185--218},

publisher = {Springer}

}

### OpenURL

### Abstract

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).

### Citations

360 |
Homotopy limits, completions and localizations
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Citation Context ...e E p,q 2 = Extq ( ˜ Hp(X), A) ⇒ πch(NZX, J[−p − q]) (see [6]). The map f induces a homology sheaf isomorphism NZX → NZY , and then a comparison of spectral sequences gives the desired result. Recall =-=[2]-=- that the category of C op -diagrams in simplicial sets has a projective model structure for which the fibrations (respectively weak equivalences) are the maps f : X → Y which are defined sectionwise ... |

200 | Sheaves in geometry and logic. A first introduction to topos theory. Corrected reprint of the 1992 edition. Universitext - Lane, Moerdijk - 1994 |

121 |
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Citation Context ...illen model structures for simplicial sheaves and simplicial presheaves on all small Grothendieck sites [11], all of which determine the same homotopy category. In the first of these structures [17], =-=[7]-=-, the monomorphisms are the cofibrations and the weak equivalences are the local weak equivalences (ie. stalkwise weak equivalences in the presence of enough stalks), and then the homotopy categories ... |

80 | Presheaves of symmetric spectra - Jardine |

46 |
Abstract homotopy theory and generalized sheaf cohomology
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Citation Context ...n the traditional sense therefore determines a picture of simplicial sheaf morphisms ∗ ← C(U) → BG. where the canonical map C(U) → ∗ is a hypercover. More generally, it has been known for a long time =-=[3]-=-, [6] that the locally fibrant simplicial sheaves (ie. simplicial sheaves which are Kan complexes in each stalk) have a partial homotopy theory which is good enough to give a calculus of fractions app... |

37 |
Simplicial objects in a Grothendieck topos
- Jardine
- 1986
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Citation Context ... traditional sense therefore determines a picture of simplicial sheaf morphisms ∗ ← C(U) → BG. where the canonical map C(U) → ∗ is a hypercover. More generally, it has been known for a long time [3], =-=[6]-=- that the locally fibrant simplicial sheaves (ie. simplicial sheaves which are Kan complexes in each stalk) have a partial homotopy theory which is good enough to give a calculus of fractions approach... |

32 | A homotopy theory for stacks
- Hollander
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Citation Context ...lly path connected in the sense that the sheaf of path components ˜π0(G) is isomorphic to the terminal sheaf. Stacks are really just homotopy types of presheaves (or sheaves) of groupoids [18], [10], =-=[5]-=-, so one may as well say that a gerbe is a locally connected presheaf of groupoids. A morphism of gerbes is a morphism G → H of presheaves of groupoids which is a weak equivalence in the sense that th... |

29 | Handbook of Categorical Algebra. Basic Category Theory - Borceux - 1994 |

22 |
Letter to A. Grothendieck
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Citation Context ... of Quillen model structures for simplicial sheaves and simplicial presheaves on all small Grothendieck sites [11], all of which determine the same homotopy category. In the first of these structures =-=[17]-=-, [7], the monomorphisms are the cofibrations and the weak equivalences are the local weak equivalences (ie. stalkwise weak equivalences in the presence of enough stalks), and then the homotopy catego... |

20 |
Strong stacks and classifying spaces, Category theory
- Joyal, Tierney
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(Show Context)
Citation Context ...hich is locally path connected in the sense that the sheaf of path components ˜π0(G) is isomorphic to the terminal sheaf. Stacks are really just homotopy types of presheaves (or sheaves) of groupoids =-=[18]-=-, [10], [5], so one may as well say that a gerbe is a locally connected presheaf of groupoids. A morphism of gerbes is a morphism G → H of presheaves of groupoids which is a weak equivalence in the se... |

15 |
Intermediate model structures for simplicial presheaves
- Jardine
(Show Context)
Citation Context ...heaves. Much has transpired in the intervening years. We now know that there is a plethora of Quillen model structures for simplicial sheaves and simplicial presheaves on all small Grothendieck sites =-=[11]-=-, all of which determine the same homotopy category. In the first of these structures [17], [7], the monomorphisms are the cofibrations and the weak equivalences are the local weak equivalences (ie. s... |

14 | Voevodsky – “A 1 -homotopy theory of schemes”, Inst - Morel, V - 1999 |

11 |
Universal Hasse-Witt classes. In Algebraic K-theory and algebraic number theory
- Jardine
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Citation Context ...f The identification of the non-abelian cohomology invariant H 1 (C, G) with morphisms [∗, BG] in the homotopy category of simplicial sheaves of Theorem 5 is, at this writing, almost twenty years old =-=[8]-=-. Unlike the original proof, the demonstration given here contains no references to hypercovers or pro objects. 2.2 Diagrams and torsors There is a local model structure for simplicial presheaves on a... |

7 |
A1-homotopy theory of schemes, Publ
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Citation Context ...model categories of simplicial presheaves, provided that those model structures are right proper (so that Theorem 2 applies). This is proved in [13]. The motivic model category of Morel and Voevodsky =-=[21]-=- is an example of a localized model structure for which this result holds. 2.3 Stack completion The overall technique displayed in this section specializes to give an explicit model for the stack asso... |

6 |
and the homotopy theory of simplicial sheaves
- Jardine, Stacks
(Show Context)
Citation Context ...−−→ GX) with morphisms [∗, holim −−−→ GX] in the homotopy category of simplicial presheaves in the statement of Theorem 7 is a consequence of Theorem 2. Theorem 7 is a generalization of Theorem 16 of =-=[10]-=-, which deals with the case where G is a sheaf of groups and X is a sheaf (ie. constant simplicial sheaf) with G-action. Example 8. Let et|S be the étale site for a scheme S, and suppose that G is a d... |

6 | Fibred sites and stack cohomology
- Jardine
- 2004
(Show Context)
Citation Context ...proved in [13], by a method which generalizes the proof of Theorem 5. This same collection of ideas is also strongly implicated in the homotopy invariance results for stack cohomology which appear in =-=[12]-=-. The definition of A-torsor and the homotopy classification result Theorem 6 have analogues in localized model categories of simplicial presheaves, provided that those model structures are right prop... |

6 |
Higher principal bundles
- Jardine, Luo
(Show Context)
Citation Context ...uivalence if it induces a weak equivalence of simplicial sets dBG → dBH. There is a natural weak equivalence of simplicial sets dBH ≃ W H relating dBH to the space of universal cocycles W H [4, V.7], =-=[16]-=-, so that G → H is a weak equivalence of 2-groupoids if and only if the induced map W G → W H is a weak equivalence of simplicial sets. One can also show that a map G → H is a weak equivalence if and ... |

5 | Diagrams and torsors
- Jardine
(Show Context)
Citation Context ...model structure for simplicial presheaves on a small site C which is Quillen equivalent to the local model structure for simplicial sheaves [7] — this has also been known for some time. Just recently =-=[13]-=-, it has been shown that both the torsor concept and the homotopy classification result Theorem 5 admit substantial generalizations, to the context of diagrams of simplicial presheaves defined on pres... |

4 |
Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids
- Luo
- 2003
(Show Context)
Citation Context ...implicial sets [4, V.7.11]. The model structure for groupoids enriched in simplicial sets restricts to a model structure for 2-groupoids, having the same definitions of fibration and weak equivalence =-=[19]-=-, and it is easy to see that both model structures are right proper. Here are some simple examples of 2-groupoids: 1) If K is a group, then there is a 2-groupoid Aut(K) with a single 0-cell, with 1-ce... |

2 | Torsors and stacks
- Jardine
- 2005
(Show Context)
Citation Context ...ucture for which this result holds. 2.3 Stack completion The overall technique displayed in this section specializes to give an explicit model for the stack associated to a sheaf of groupoids H — see =-=[14]-=-. In general, the stack associated to H has global sections with objects given by the “discrete” H-torsors. A discrete H-torsor is a H-torsor X as above, with the extra requirement that X is simplicia... |

1 | The homotopy classification of gerbes
- Jardine
- 2006
(Show Context)
Citation Context ...n Section 5 we discuss but do not prove a classification theorem for gerbes up to local equivalence as path components of a suitable cocycle category. This last result is Theorem 13 — it is proved in =-=[15]-=-. The cocycles appearing in the proofs of Theorems 5, 12 and 13 are canonically defined. 1 Cocycles Suppose that M is a right proper closed model category, and suppose further that that the weak equiv... |