## Homotopical algebraic geometry. II. Geometric stacks and applications (2006)

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Citations: | 22 - 12 self |

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@MISC{Toën06homotopicalalgebraic,

author = {Bertrand Toën and Gabriele Vezzosi},

title = {Homotopical algebraic geometry. II. Geometric stacks and applications},

year = {2006}

}

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480 | Étale cohomology
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Citation Context ...s h ≃ h ≃ Rh : k − Aff −→ Ho(k − Aff ∧ ), which is nothing else than the natural composition k − Aff h �� ∧ k − Aff �� ∧ Ho(k − Aff ). We let τ = ét, the usual étale pre-topology on k − Aff (see e.g. =-=[Mil]-=-). Recall that a family of morphisms {Xi = Spec Ai −→ X = Spec A}i∈I 129130 2.1. GEOMETRIC n-STACKS IN ALGEBRAIC GEOMETRY (AFTER C. SIMPSON) is an ét-covering family if and only if it contains a fini... |

448 | Théorie des topos et cohomologie étale des schémas - Artin, Grothendieck, et al. - 1977 |

326 |
Higher algebraic K-theory. I
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Citation Context ...icular, it only remains to show that the adjunction morphism Id −→ RSLN is an isomorphism. But this last assumption follows from the definition the functor N and from a standard lemma (se for example =-=[Q3]-=-), which shows that the homotopy fiber at x ∈ C of N(F) −→ N(C) is naturally equivalent to F(x) when F is fibrant in (C, C) ∧ . □ Recall from [HAGI, Lem. 4.2.2] that for any object x ∈ C, one can cons... |

325 | Homotopical algebra - Quillen - 1967 |

285 |
Éléments de géométrie algébrique
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Citation Context ...orphism U −→ F, which is seen to be smooth using Cor. 2.2.5.3. Using this lifting of smooth morphisms from it0 to F, one produces an n-atlas for F by lifting an n-atlas of it0(F). □ �� FBibliography =-=[EGAI]-=- A. Grothendieck, J. Dieudonné, Eléments de Géométrie Algébrique, I, Springer-Verlag, New York 1971. [AHS] M. Ando, M. J. Hopkins, N. P. Strickland, Elliptic spectra, the Witten genus, and the theorem... |

279 |
Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas
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Citation Context ...ine schemes. But, passing to t0 we find a smooth covering family {t0(Ui) −→ t0(X)}, such that each morphism t0(Ui) −→ t0(Y ) is smooth, and�� �� 140 2.2. DERIVED ALGEBRAIC GEOMETRY we know (see e.g. =-=[EGAIV]-=-) that this implies that t0(X) −→ t0(Y ) is a smooth morphism of affine schemes. Finally, property (4) of 1.3.2.11 is obvious. □ We have verified our assumptions 1.3.2.2 and 1.3.2.11 for the étale mod... |

267 | Introduction to homological algebra - Weibel - 1994 |

189 | Symmetric spectra
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Citation Context ... Two other context of applications are the cases when C = C(k), the model category of unbounded complexes over some commutative Q-algebra k, and when C = Sp Σ the model category of symmetric spectra (=-=[HSS]-=-). The first of this context gives rise to what we call complicial algebraic geometry, an unbounded version of derived algebraic geometry. It turns out that linear algebra over C(k) behaves rather dif... |

169 | Complexe cotangent et déformations. I - Illusie - 1971 |

164 | Simplicial homotopy theory
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Citation Context ...ategory of Usimplicial k-modules, endowed with the levelwise tensor product and the usual model structure for which equivalences and fibrations are defined on the underlying simplicial sets (see e.g. =-=[Goe-Ja]-=-). The category Comm(C) is then the category of simplicial commutative k-algebras, and for A ∈ sModk, A − Mod is the category of simplicial modules over the simplicial ring A. Assumptions 1.1.0.1, 1.1... |

143 |
Model categories, Mathematical surveys and monographs
- Hovey
- 1998
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Citation Context ...gebraic geometry, a global counter part of the derived deformation theory of V. Drinfel’d, M. Kontsevich and al. The setting. Our basic datum is a symmetric monoidal model category C (in the sense of =-=[Ho1]-=-), on which certain conditions are imposed (see assumptions 1.1.0.1, 1.1.0.2, 1.1.0.3 and 1.1.0.4). We briefly discuss these requirements here. The model category C is assumed to satisfy some reasonab... |

143 |
modules, and algebras in stable homotopy theory
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- 1997
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Citation Context ...∆ni+1 ] It is not difficult to check that an object A ∈ sk − Alg if finitely presented in the sense of Def. 3.3.1 if and only if it is equivalent to a retract of a finite cell object (see for example =-=[EKMM]-=-). More 109generally, for A ∈ sk − Alg, there exists a notion of finite cell object in A − Algs using the elementary morphisms A[S n ] := A ⊗k k[S n ] −→ A[∆ n+1 ] := A ⊗k k[∆ n+1 ]. In the same way,... |

134 | Revetements etales et groupe fondamental - Grothendieck - 1971 |

123 | Catégories tannakiennes. In: The Grothendieck Festschrift - Deligne - 1990 |

121 | Les schémas de modules de courbes elliptiques” in Modular Functions of One Variable - DELIGNE, RAPOPORT - 1972 |

114 | Simplicial presheaves - Jardine - 1987 |

112 | Deformations of algebras over operads and the Deligne conjecture. Conférence Moshé Flato
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Citation Context ...a brave new version of Galois theory, including brave new notions of étaleness which are very close to our notions (see [Ro]). A construction of the moduli of dg-algebras and dg-categories appears in =-=[Ko-So]-=-. These moduli are only formal moduli by construction, and we propose our D-stacks Ass and Cat∗ as their global geometrical counterparts. We wish to mention the work of M. Spitzweck [Sp], in which he ... |

112 | Model categories of diagram spectra - Mandell, May, et al. |

105 | Le groupe fondamental de la droite projective moins trois points - Deligne - 1999 |

103 |
Champs Algébriques
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- 2000
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Citation Context ...metric stack has an obstruction theory (Thm. 1.4.3.2). This last result can be considered as a far reaching generalization of the exixtence of cotangent complexes for algebraic stacks as presented in =-=[La-Mo]-=-. Higher Artin stacks (after C. Simpson). As a first example of application, we show how our general notion of geometric stacks specializes to C. Simpson’s algebraic n-stacks introduced in [S3]. For t... |

76 | Stable model categories are categories of modules, Topology 42
- Shipley, Schwede
- 2003
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Citation Context ...nce for all our base model category as well as several general assumptions it will satisfy. All along this paragraph, we refer to [Ho1] for the general definition of monoidal model categories, and to =-=[Schw-Shi]-=- for general results about monoids and modules in monoidal model categories. From now on, and all along this work, we fix three universes U ∈ V ∈ W. We also let (C, ⊗,1) be a symmetric monoidal model ... |

64 |
Simplicial localization of categories
- Dwyer, Kan
- 1980
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Citation Context ...nt objects in M f , to SSet TorsX∗ : (M f ) op −→ SSet. This functor sends equivalences in M f to equivalences of simplicial sets, and therefore induces a Ho(SSet)-enriched functor (using for example =-=[D-K1]-=-) TorsX∗ : Ho(M f ) op ≃ Ho(M) op −→ Ho(SSet). In other words, one has natural morphisms in Ho(SSet) MapM(F,F ′ ) −→ MapSSet(TorsX∗(F ′ ),TorsX∗(F)), compatible with compositions. The main classificat... |

63 |
P.: Groupes Algébriques
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Citation Context ... reconstructed as certain Zariski sheaves on the opposite category of commutative rings, which are obtained by gluing affine schemes via Zariski open immersions (see for example the first chapters of =-=[Dem-Gab]-=-). The fact that the notion of schemes has such a purely categorical interpretation has naturally lead to the theory of relative algebraic geometry, in which the base symmetric monoidal category Z − M... |

54 | Spectra and symmetric spectra in general model categories
- Hovey
(Show Context)
Citation Context ...A ⊗A − : A − Mod −→ A − Mod HomA (S 1 A , −) : A − Mod −→ A − Mod. Furthermore, assumption 1.1.0.2 implies that S1 A ⊗A − is a left Quillen functor. We can therefore apply the general construction of =-=[Ho2]-=- in order to produce a model category SpS1 A(A − Mod), of spectra in A − Mod with respect to the left Quillen endofunctor S1 A ⊗A −.40 1.2. PRELIMINARIES ON LINEAR AND COMMUTATIVE ALGEBRA IN AN HA CO... |

54 | André-Quillen cohomology of commutative S-algebras
- Basterra
- 1999
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Citation Context ...initions we give are the right one, but simply that they exist and are well behaved. 173.1 Derivations and the cotangent complex This paragraph is nothing else than a rewriting of the first pages of =-=[Ba]-=-, that stay valid in our general context. Let A ∈ Comm(C) be a commutative monoid in C, and M be an A-module. We define a new commutative monoid A ⊕ M in the following way. The underlying object of A ... |

53 | The Hodge filtration on nonabelian cohomology. Algebraic geometry - Simpson - 1995 |

50 | Combinatorial model categories have presentations
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Citation Context ...e monoidal structure). These 4assumptions are only made for the convenience of certain constructions, and might clearly be omitted. The model category C is also assumed to be combinatorial (see e.g. =-=[Du2]-=-), making it reasonably behaved with respect to localizations techniques. The first really important assumption on C states that it is pointed (i.e. that the final and initial object coincides) and th... |

39 | A convenient model category for commutative ring spectra, from: “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic
- Shipley
- 2004
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Citation Context ...ory of co-simplicial k-modules, our assumptions would again be satisfied. 4. Let Sp Σ be the category of U-symmetric spectra and its smash product, endowed with the positive stable model structure of =-=[Shi]-=-. Then, Comm(C) is the category of commutative symmetric ring spectra, and assumptions 2.0.1, 2.0.2, 2.0.3 and 2.0.4 are known to be satisfied (see [Shi, Thm. 3.1, Thm. 3.2, Cor. 4.3]). It is importan... |

37 | Universal homotopy theories - Dugger |

35 |
Homotopy commutative diagrams and their realizations
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Citation Context ...sub-category of equivalences in A −Mod. In particular, π0(QCoh(A)) is in bijection with isomorphisms classes of objects in Ho(A − Mod) (i.e. equivalent classes of objects in A − Mod). Furthermore, by =-=[D-K3]-=-, for any x ∈ QCoh(A), corresponding to an equivalence class of M ∈ A − Mod, the connected component of QCoh(A) containing x is naturally equivalent to BAut(M), where Aut(M) is the simplicial monoid o... |

33 |
Model Categories and their
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Citation Context ...at for any two objects x and y in C the set HomC(x, y) is U-small. In the same way, a category C is U-small is it is isomorphic to some element in U. Our references for model categories are [Ho1] and =-=[Hi]-=-. By definition, our model categories will always be closed model categories, will have all small limits and colimits and the functorial factorization property. The word equivalence will always mean w... |

31 | Segal topoi and stacks over Segal categories. Available for download: math.AG/0212330 - Toën, Vezzosi |

30 | Model categories and more general abstract homotopy theory, Book in preparation, available at http://www-math.mit.edu/ ∼ psh - Dwyer, Hirschhorn, et al. |

30 |
Une structure de catégorie de modèles de Quillen sur la catégorie des dgcatégories, Comptes Rendus de l’Acadmie de Sciences de Paris 340
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Citation Context ...) → z in π0(E), there exists an isomorphism u : x → y in π0(D) such that π0(f)(u) = v. With these notions of fibrations and equivalences, the category A − dgCat is a model category. This is proved in =-=[Tab]-=- when A is a commutative ring. The general case of categories enriched in a well behaved monoidal model category has been worked out recently by J. Tapia (private communication). For A ∈ k−cdga, we de... |

29 | Homotopical algebraic Geometry I: Topos Theory
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- 2005
(Show Context)
Citation Context ...algebra in an abstract monoidal model category, such as derivations, étale and smooth morphisms, flat and projective modules, etc. We then use our theory of stacks over model categories introduced in =-=[HAGI]-=- in order to define a general notion of geometric stacks over a base symmetric monoidal model category C, and prove that this notion satisfies expected properties. The rest of the paper consists in sp... |

27 | Equivalences between homotopy theories of diagrams, Algebraic topology and algebraic K-theory - Dwyer, Kan - 1987 |

27 | Algebraic geometry over model categories. A general approach to Derived Algebraic Geometry - Toën, Vezzosi - 2013 |

26 |
On the (co-)homology of commutative rings, Applications of Categorical Algebra
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Citation Context ...A − Mod. For any morphism of simplicial commutative k-algebras A −→ B, the B-module L B/A constructed in 1.2.1.2 is naturally isomorphic in Ho(B − Mod) to D. Quillen’s cotangent complex introduced in =-=[Q2]-=-. In particular, if A −→ B is a morphism 123�� � �� 124 2.2. DERIVED ALGEBRAIC GEOMETRY between (non-simplicial) commutative k-algebras, then we have π0(L B/A) ≃ Ω 1 B/A . More generally, we find by ... |

25 |
Spaces of algebra structures and cohomology of operads
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Citation Context ...k n )) is defined as sk − Alg −→ SSet A ↦→ Map(O,End(k n ))(A) := HomOp(QO,End(A n )), where Hom Op denotes the simplicial set of morphism in the simplicial category of operads in sk−Mod. As we said, =-=[Re]-=- implies that ÃlgO n is isomorphic to Map(O,End(k n )). Therefore, it remain to show that Map(O,End(k n )) is a representable D − -stack. For this, we can write O, up to an equivalence, as the homotop... |

23 | Brave New” algebraic geometry and global derived moduli spaces of ring spectra - Toën, Vezzosi - 2003 |

22 |
J.P.May “Operads, algebras, modules and motives.” Astrisque No
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Citation Context ...g A −→ A ′ such that M ′ := M ⊗ L A A′ is a perfect A ′ -module, and we need to prove that M is itself perfect. For this we use that perfect A-modules are precisely the finitely presented object (see =-=[Kr-Ma]-=-). Let us write M as a filtered colimit of finitely presented objects M ≃ HocolimiEi. As M ′ is finitely presented we see that there is an i such that the morphism u : Ei −→ M admits a retraction afte... |

19 |
André-Quillen (co)homology for simplicial algebras over simplicial operads, Une Dégustation Topologique [Topological Morsels]: Homotopy Theory
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Citation Context ...nes a well defined morphism of stacks B : ∗ −→ Alg O n . We are going to describe the cotangent complex of Alg O n at B using the notion of (derived) derivations for O-algebras. For this, recall from =-=[Goe-Hop]-=- the notion of O-derivations from B and with coefficients in a B-module. For any B-module M, one can define the square zero extension B ⊕ M of B by M, which is another O-algebra together with a natura... |

19 | Stacks and the homotopy theory of simplicial sheaves, in Equivariant stable homotopy theory and related areas - Jardine - 2000 |

19 | Strong stacks and classifying spaces, Category theory - Joyal, Tierney - 1990 |

19 | Vers une interprétation Galoisienne de la théorie de l’homotopie, Cahiers de top. et geom. diff. cat - Toën - 2002 |

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17 |
Homotopy over the complex numbers and generalized cohomology theory, in Moduli of vector bundles (Taniguchi Symposium
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Citation Context ...1], where TY is the tangent sheaf of Y −→ Speck. Let us now suppose that k = C is the field of complex numbers, and let X be a smooth and projective variety. We will be interested in the sheaf XDR of =-=[S1]-=-, defined by XDR(A) := Ared, for a commutative C-algebra A. Recall that the stack MDR(X) is defined as the stack of morphisms from XDR to Vectn, and is identified with the stack of flat bundles on X (... |

17 | Homotopical and higher categorical structures in algebraic geometry, Hablitation Thesis available at math.AG/0312262 - Toën |

16 | Injective resolutions of BG and derived moduli spaces of local systems - Kapranov |

16 | Picard groups, Grothendieck rings, and Burnside rings of categories
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(Show Context)
Citation Context ... objects are stable by retracts in Ho(M). Proof. (1), (2) and (3) are standard, as perfect objects are precisely the strongly dualizable objects of the closed monoidal category Ho(M) (see for example =-=[May2]-=-). (4) Let x be a perfect object in M, and {zi}i∈I be a filtered diagram of objects in M. Let x ∨ := RHom(x,1) the dual of x in Ho(M). Then, we have MapM(x, Hocolimizi) ≃ MapM(1, x ∨ ⊗ L Hocolimizi) ≃... |