## Fibred sites and stack cohomology

Venue: | Math. Z |

Citations: | 6 - 6 self |

### BibTeX

@ARTICLE{Jardine_fibredsites,

author = {J. F. Jardine},

title = {Fibred sites and stack cohomology},

journal = {Math. Z},

year = {},

pages = {2006}

}

### OpenURL

### Abstract

A stack G is traditionally defined to be a pseudofunctor on a Grothendieck

### Citations

326 |
Higher algebraic K-theory. I
- Quillen
- 1973
(Show Context)
Citation Context ... derived functor Lf ∗ : Ho(sSet G ) → Ho(sSet H ) which is an inverse up to natural isomorphism for the derived restriction functor Rf∗ : Ho(sSet H ) → Ho(sSet G ). Here’s a result that is well known =-=[18]-=-, but stated and proved in a completely functorial manner. We will need the functoriality for a corresponding result on presheaves of categories which will be used in the next section of this paper. L... |

307 |
Homotopy limits, completions and localizations
- Bousfield, Kan
- 1972
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Citation Context ...bred site C/A is actually a type of coarse equivariant theory of A op -diagrams — one says “coarse” because this is an enriched version of the old Bousfield-Kan theory for diagrams of simplicial sets =-=[1]-=-. In the case when A is a presheaf of groupoids G, this assignment of homotopy colimits determines an equivalence of homotopy categories Ho(s Pre(C/G)) ≃ Ho(s Pre(C)/BG op ) (1) which generalizes the ... |

236 | Categories for the working mathematician. Graduate Texts - Lane - 1998 |

191 | Symmetric spectra - Hovey, Shipley, et al. |

127 | Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete 39 - Laumon, Moret-Bailly - 2000 |

114 | Cohomologie non abélienne - GIRAUD - 1971 |

114 | Simplicial presheaves - Jardine - 1987 |

75 | Motivic symmetric spectra
- Jardine
- 2000
(Show Context)
Citation Context ...poids G the restriction functor ψ∗ : Spt Σ (C/G) → Spt Σ (C/ Ob(G)) between the respective categories of presheaves of symmetric spectra preserves stable fibrations and trivial stable fibrations (see =-=[10]-=-). It follows that its left adjoint ψ ∗ preserves cofibrations and trivial cofibrations. Say that a map p : X → Y of symmetric spectra on C/G is a projective fibration if the induced map p∗ : ψ∗X → ψ∗... |

35 |
Simplicial objects in a Grothendieck topos, in Applications of algebraic K-theory to algebraic geometry and number theory
- Jardine
- 1986
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Citation Context ... Z) ≃ −→ hom(1X, Z), where the function space hom(1X, Z) is a homotopy inverse limit of the simplicial sets Γ∗Zn, computed on the respective sites C/Xn. This is, effectively, an old observation — see =-=[6]-=-. 6�� � Example 4. Suppose that J is a small category, and identify J with a constant presheaf of categories on C. The category C/J has, for objects, all pairs (U, x) where U is an object of C and x ... |

29 | A Homotopy Theory for Stacks
- Hollander
(Show Context)
Citation Context ...nt simplicial presheaf. This description of stacks was a major conceptual breakthrough which was initiated by Joyal and Tierney [14] in the case of sheaves of groupoids and was completed by Hollander =-=[4]-=- for presheaves of groupoids. Stack completion becomes a fibrant model in this setup, and it is now well understood that path components (or isomorphism classes) in the global sections of a stack G ar... |

25 |
Generalized Etale Cohomology Theories
- Jardine
- 1997
(Show Context)
Citation Context ...Ob(G). By definition, f is a projective fibration if and only if f∗ is a global fibration on C/ Ob(G). We also know, from Lemma 15, that the restriction functor ψ∗ preserves global fibrations. Recall =-=[9]-=- that a map g : Z → W of presheaves of spectra is a stable equivalence if the induced map QJX → QJY is a levelwise weak equivalence, where 32�� �� � � �� �� X → JX is a natural choice of strictly fib... |

19 |
Strong stacks and classifying spaces, Category theory
- Joyal, Tierney
- 1990
(Show Context)
Citation Context ...resheaves on C. Thus, G is a stack if and only if BG is a globally fibrant simplicial presheaf. This description of stacks was a major conceptual breakthrough which was initiated by Joyal and Tierney =-=[14]-=- in the case of sheaves of groupoids and was completed by Hollander [4] for presheaves of groupoids. Stack completion becomes a fibrant model in this setup, and it is now well understood that path com... |

10 |
Universal Hasse-Witt classes. In Algebraic K-theory and algebraic number theory
- Jardine
- 1987
(Show Context)
Citation Context ...cial presheaves. This gives a rather striking generalization of the early result that identified the homotopy invariants [∗, BH] arising from sheaves of groups H with isomorphism classes of H-torsors =-=[8]-=-. We also now understand what the 1higher order analogues of H-torsors should be, and a homotopy theoretic (and geometric) identification of these higher order torsors has been achieved [13]. This pa... |

6 | and the homotopy theory of simplicial sheaves - Jardine, Stacks |

5 | On the homotopy theory of sheaves of simplicial groupoids - Joyal, Tierney - 1996 |

3 |
Higher order principal bundles
- Jardine, Luo
- 1993
(Show Context)
Citation Context ...H-torsors [8]. We also now understand what the 1higher order analogues of H-torsors should be, and a homotopy theoretic (and geometric) identification of these higher order torsors has been achieved =-=[13]-=-. This paper brings stack cohomology into this arena, by giving an homotopy theoretic description of the invariant in terms of presheaves of groupoids. One of the more important consequences of this a... |