Hypercovers and simplicial presheaves

Dugger, Daniel; Hollander, Sharon; Isaksen, Daniel C.

doi:10.1017/s0305004103007175

Cited by 159 publications

(222 citation statements)

References 22 publications

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“…Fibrancy is shown to be a local condition in [DHI04], and hence follows from Lemma 8.3.2 and the fibrancy of the presheaves E(K p ) of Section 8.3.…”

Section: Descent From Compact Open Subgroupsmentioning

confidence: 85%

“…Then there exists a presheaf of E ∞ -ring spectra E G on theétale site of X ∧ mA , such that (1) E G satisfies homotopy descent: it is (locally) fibrant in the Jardine model structure [Jar00], [DHI04]. (2) For every formal affineétale open f : Spf(R) → X ∧ mA , the E ∞ -ring spectrum of sections E G (R) is weakly even periodic, with E G (R) 0 = R. There is an isomorphism…”

Section: Goerss and Hopkins [Gh04] Extended Work Of Hopkins And Millermentioning

confidence: 99%

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Topological automorphic forms

Behrens¹,

Lawson²

2010

Memoirs of the AMS

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Introduction ix 0.1. Background and motivation ix 0.2. Subject matter of this book xvii 0.3. Organization of this book xxiii 0.4. Acknowledgments xxv Chapter 1. p-divisible groups 1.1. Definitions 1.2. Classification Chapter 2. The Honda-Tate classification 2.1. Abelian varieties over finite fields 2.2. Abelian varieties over F p Chapter 3. Tate modules and level structures 3.1. Tate modules of abelian varieties 3.2. Virtual subgroups and quasi-isogenies 3.3. Level structures 3.4. The Tate representation 3.5. Homomorphisms of abelian schemes Chapter 4. Polarizations 4.1. Polarizations 4.2. The Rosati involution 4.3. The Weil pairing 4.4. Polarizations of B-linear abelian varieties 4.5. Induced polarizations 4.6. Classification of weak polarizations Chapter 5. Forms and involutions 5.1. Hermitian forms 5.2. Unitary and similitude groups 5.3. Classification of forms Chapter 6. Shimura varieties of type U (1, n − 1) 6.1. Motivation 6.2. Initial data 6.3. Statement of the moduli problem 6.4. Equivalence of the moduli problems 6.5. Moduli problems with level structure 6.6. Shimura stacks v vi CONTENTS Chapter 7. Deformation theory 7.1. Deformations of p-divisible groups 7.2. Serre-Tate theory 7.3. Deformation theory of points of Sh Chapter 8. Topological automorphic forms 8.1. The generalized Hopkins-Miller theorem 8.2. The descent spectral sequence 8.3. Application to Shimura stacks Chapter 9. Relationship to automorphic forms 9.1. Alternate description of Sh(K p ) 9.2. Description of Sh(K p ) F 9.3. Description of Sh(K p ) C 9.4. Automorphic forms Chapter 10. Smooth G-spectra 10.1. Smooth G-sets 10.2. The category of simplicial smooth G-sets 10.3. The category of smooth G-spectra 10.4. Smooth homotopy fixed points 10.5. Restriction, induction, and coinduction 10.6. Descent from compact open subgroups 10.7. Transfer maps and the Burnside category Chapter 11. Operations on TAF 11.1. The E ∞ -action of GU (A p,∞ ) 11.2. Hecke operators Chapter 12. Buildings 12.1. Terminology 12.2. The buildings for GL and SL 12.3. The buildings for U and GU Chapter 13. Hypercohomology of adele groups 13.1. Definition of Q GU and Q U 13.2. The semi-cosimplicial resolution Chapter 14. K(n)-local theory 14.1. Endomorphisms of mod p points 14.2. Approximation results 14.3. The height n locus of Sh(K p ) 14.4. K(n)-local TAF 14.5. K(n)-local Q U Chapter 15. Example: chromatic level 1 15.1. Unit groups and the K(1)-local sphere 15.2. Topological automorphic forms in chromatic filtration 1 Bibliography Index

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“…Fibrancy is shown to be a local condition in [DHI04], and hence follows from Lemma 8.3.2 and the fibrancy of the presheaves E(K p ) of Section 8.3.…”

Section: Descent From Compact Open Subgroupsmentioning

confidence: 85%

Section: Goerss and Hopkins [Gh04] Extended Work Of Hopkins And Millermentioning

confidence: 99%

Topological automorphic forms

Behrens¹,

Lawson²

2010

Memoirs of the AMS

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“…[DHI,Proposition 3.1] A map F → G between locally fibrant simplicial presheaves is a weak equivalence if and only if given a commutative square…”

Section: Generalized Artin Stacksmentioning

confidence: 99%

Characterizing Artin stacks

Hollander¹

2010

Math. Z.

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Abstract. We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site C. There is a natural method for extending a property P of morphisms of sheaves on C to a property P of morphisms of presheaves of groupoids. We prove that the property P is homotopy invariant in the local model structure on P (C, Grpd) L when P is stable under pullback and local on the target. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, we obtain homotopy theoretic characterizations of algebraic and Artin stacks as those which are equivalent to simplicial objects in C satisfying certain analogues of the Kan conditions.The definition of Artin stack can naturally be placed within a hierarchy which roughly measures how far a stack is from being representable. We call the higher analogues of Artin stacks n-algebraic stacks, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, n-algebraic is the same as 3-algebraic for all n ≥ 3.As an application of these results we show that a stack is n-algebraic if and only if the homotopy orbits of a group action on it is.

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“…Notons ensuite qu'un morphisme (étale) surjectif de schémas f : X → Y dans Sm/S induit une application surjective X(C) → Y (C) : si y ∈ Y (C), la fibre X y de f au-dessus de y est un schéma de type fini sur C et non vide, il possède un C-point (10) . Il en résulte que le foncteur ι −1 X est continu ; comme il commute aux limites projectives finies, on a bien un morphisme de sites (cf.…”

Section: L'application Raisonnableunclassified

“…[10] et [11]). Enfin, dans la section 6, on propose une définition d'une variante « naïve » de la catégorie SH T (S , I) : admettant une description très simple à partie de la catégorie homotopique pointée H • (S , I), cette catégorie s'avère être équivalente au quotient de SH T (S , I) par l'idéal de morphismes (de carré nul) constitué par les morphismes dits « stablement fantômes ».…”

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Catégorie homotopique stable d’un site suspendu avec intervalle

Riou¹

2007

Bul. Soc. Math. France

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Résumé. -Cet article présente la construction de la catégorie homotopique stable d'un site suspendu avec intervalle arbitraire. La fonctorialité de cette construction est étudiée, avec des applications à la théorie homotopique des schémas introduite par F. Morel et V. Voevodsky. Abstract (The stable homotopy category of an hanging site with interval)This article describes the construction of the stable homotopy category of an arbitrary hanging site with interval. The functoriality of this construction is studied and has applications to the A 1 -homotopy theory introduced by F. Morel and V. Voevodsky.

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Hypercovers and simplicial presheaves

Cited by 159 publications

References 22 publications

Topological automorphic forms

Topological automorphic forms

Characterizing Artin stacks

Catégorie homotopique stable d’un site suspendu avec intervalle

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