### BibTeX

@MISC{Lurie06highertopos,

author = {Jacob Lurie},

title = {Higher topos theory},

year = {2006}

}

### OpenURL

### Abstract

Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of G-valued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category

### Citations

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Citation Context ...egory C ′ ⋆ G C ′′ , then C ⋆ G◦F C ′′ is obtained by forming the pushout (C ⋆ F C ′ ) ∐ C ′ (C ′ ⋆ G C ′′ ) and then discarding the objects of C ′ . Now, giving a category equipped with a functor to =-=[2]-=- is equivalent to giving a triple of categories C, C ′ , C ′′ , together with correspondences F ∈ M(C, C ′ ), G ∈ M(C ′ , C ′′ ), H ∈ M(C, C ′′ ) and a map α : G ◦ F → H. But the map α need not be an ... |

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Citation Context ...pace, and 0 ≤ n ≤ ∞. We can extract an n-category π≤nX (roughly) as follows. The objects of π≤nX are the points of X. If x, y ∈ X, then the morphisms from x to y in π≤nX are given by continuous paths =-=[0, 1]-=- → X starting at x and ending at y. The 2-morphisms are given by homotopies of paths, the 3-morphisms by homotopies between homotopies, and so forth. Finally, if n < ∞, then two n-morphisms of π≤nX ar... |

222 | Model Categories
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Citation Context ...nce of model structures on diagram categories. Our exposition is rather dense; for a more leisurely account of the theory of model categories we refer the reader to one of the standard texts, such as =-=[27]-=-. 540�� � A.1 Category Theory Familiarity with classical category theory is the main prerequisite for reading this book. In this section, we will fix some of the notation that we will use when discus... |

196 |
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Citation Context ...elonging to B, the induced map SingX(Y, U)• SingX(Y, U)• is a Kan fibration. → 473Proof. The proof uses the theory of cofibrantly generated model categories; we give a sketch and refer the reader to =-=[26]-=- for more details. Say that a morphism Y → Z in Top /X is a cofibration if it has the left lifting property with respect to every trivial fibration in Top /X . We begin by observing that a map Y → Z i... |

183 | Simplicial homotopy theory
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Citation Context ...al sets. We will assume that the reader has some familiarity with the theory of simplicial sets; a brief review of this theory is included in §A.2.7, and a more extensive introduction can be found in =-=[21]-=-. The theory of simplicial sets originated as a combinatorial approach to homotopy theory. Given any topological space X, one can associated a simplicial set Sing X, whose n-simplices are precisely th... |

165 |
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(Show Context)
Citation Context ...let X be the topos of étale sheaves on X, and let X be the associated 1-localic ∞-topos. The shape Sh(X) defined above is closely related to the étale homotopy type introduced by Artin and Mazur (see =-=[3]-=-). There are three important differences: (1) Artin and Mazur work with pro-objects in the homotopy category H, rather than with actual proobjects of S. (2) The étale homotopy type of [3] is construct... |

161 |
Spectral Theory and Analytic Geometry over Non-Archimedean Fields
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(Show Context)
Citation Context ...that the global sections functor g∗ : Shv(XZR) → Shv(∗) is proper, which follows from Corollary 7.3.5.3. We now observe that the topological space XB is paracompact and has finite covering dimension (=-=[4]-=-, Corollary 3.2.8), so that Shv(XB) has enough points (Corollary 7.2.1.24). According to Proposition 7.3.6.4, it suffices to show that for every fiber diagram X ′ �� Shv(XZR) Shv(∗) q∗ � Shv(XB), the ... |

129 |
Resolutions of unbounded complexes
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Citation Context ...ase change theorem. We refer the reader to §7.3 for a precise statement and proof. Remark 6.5.4.2. A similar issue arises in classical sheaf theory if one chooses to work with unbounded complexes. In =-=[46]-=-, Spaltenstein defines a derived category of unbounded complexes of sheaves on X, where X is a topological space. His definition forces all quasi-isomorphisms to become invertible, which is analogous ... |

121 |
Simplicial presheaves
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Citation Context ...laid out by Grothendieck in his vision of a theory of higher stacks. This vision has subsequently been realized in the work of various authors (most notably Brown, Joyal, and Jardine; see for example =-=[28]-=-), who employ various formalisms based on simplicial (pre)sheaves on X. The resulting theories are essentially equivalent, and we shall refer to them collectively as the Brown-Joyal-Jardine theory. Ac... |

118 |
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Citation Context ...pplies also when A is a non-abelian coefficient system. The appropriate definition requires the vanishing of cohomology for coefficient systems which are defined only up to inner automorphisms, as in =-=[20]-=-. With the appropriate modifications, Theorem 7.2.2.29 below remains valid for n < 2. The cohomological dimension of an ∞-topos X is closely related to the homotopy dimension of X. If X has homotopy d... |

114 | Coherence for tricategories - Gordon, Power, et al. - 1995 |

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Citation Context ...ing algebra is a Heyting space X having dimension at most equal to the dimension of V . The results of this section therefore apply to X. More generally, let T be an o-minimal theory (see for example =-=[52]-=-), and let Sn denote the set of complete n-types of T. We endow Sn with the topology generated by the sets Uφ = {p : φ ∈ p}, where φ ranges over formula with n free variables such that the openness of... |

52 | Combinatorial model categories have presentations
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(Show Context)
Citation Context ...(2). Moreover, it also shows that (2) ⇒ (1) in the special case where A is a localization of a category of simplicial presheaves. We now complete the proof by invoking the following result, proven in =-=[13]-=-: for every combinatorial model category A, there exists a small category D, a set S of morphisms of Set∆ Dop , and a Quillen equivalence of A with S −1 Set D ∆. Moreover, the proof given in [13] show... |

51 | Etale Homotopy of Simplicial Schemes - Friedlander - 1982 |

48 | A model category structure on the category of simplicial categories
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Citation Context ...ving a review of the theory of simplicial categories. We will begin in §A.3.1 by introducing a model structure on the category Cat∆ of (small) simplicial categories, which was constructed by Bergner (=-=[6]-=-). Putting aside set-theoretic technicalities, every simplicial model category A provides a fibrant object A ◦ of Cat∆. In §A.3.2, we will introduce a path space object for A ◦ , which will allow us t... |

48 | Cech and Steenrod homotopy theory with applications to algebraic topology - Edwards, Hastings - 1976 |

47 | Abstract homotopy theory and generalized sheaf cohomology - Brown |

47 | Etale Cohomology and the Weil Conjecture - Freitag, Kiehl - 1988 |

45 |
Theory of quasi-categories
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Citation Context ...ndowed with a model structure for which the fibrant objects are precisely the ∞-categories. The original construction of this model structure is due to Joyal, who uses purely combinatorial arguments (=-=[31]-=-). In this section, we will exploit the relationship between simplicial categories and ∞-categories to give an alternative description of this model structure. Our discussion will make use of a model ... |

44 | der Put, Rigid Analytic Geometry and its Applications, Birkhauser - Fresnel, van - 2004 |

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Citation Context ... and let C0 denote the (ordinary) category of finitely generated free R-modules. Then C = Ind(C0) is equivalent to the category of flat R-modules (by Lazard’s theorem; see for example the appendix of =-=[33]-=-). The compact objects of C are precisely the finitely generated projective R-modules, which need not be free. Nevertheless, the naive guess is not far off, in virtue of the following result: Lemma 5.... |

36 |
The localization of spaces with respect to homology. Topology 14
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(Show Context)
Citation Context ...there is a smallest saturated collection containing S: this permits us to define a localization S −1 C ⊆ C. The ideas presented in this section go back (at least) to Bousfield; we refer the reader to =-=[9]-=- for a discussion in a more classical setting. Definition 5.5.4.1. Let C be an ∞-category and S a collection of morphisms of C. We say that an object Z of C is S-local if, for every morphism s : X → Y... |

35 |
Quasi-categories and Kan complexes
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(Show Context)
Citation Context ...hen one wishes to perform even simple categorical constructions. As a remedy, we will introduce the more convenient formalism of ∞-categories (called weak Kan complexes in [7] and quasi-categories in =-=[30]-=-), which provides a more suitable setting for adaptations of sophisticated category-theoretic ideas. Our goal in §1.1.1 is to introduce both approaches and to explain why they are equivalent to one an... |

35 | Model categories and more general abstract homotopy theory, preprint available at www-math.mit.edu/~psh/#Mom - Dwyer, Hirschhorn, et al. |

33 | Hypercovers and simplicial presheaves
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(Show Context)
Citation Context ...ompletion X ∧ . In §6.5.3, we will show that an ∞-topos X is hypercomplete if and only if X satisfies a descent condition with respect to hypercoverings (other versions of this result can be found in =-=[14]-=- and [51]). Carefully distinguishing between an ∞-topos X and its hypercompletion X ∧ is the key to solving the problem described in §. The Brown-Joyal-Jardine theory of simplicial (pre)sheaves on a t... |

31 |
Segal topoi and stacks over Segal categories. Available for download: math.AG/0212330
- Toën, Vezzosi
(Show Context)
Citation Context ...ated to this model category is an ∞-topos in our sense. This construction is generalized from ordinary categories with a Grothendieck topology to simplicial categories with a Grothendieck topology in =-=[51]-=- (and again produces ∞-topoi). However, not every ∞-topos arises in this way: one can construct only ∞-topoi which are hypercomplete (called t-complete in [51]); we will summarize the situation in Sec... |

30 |
Proper maps of toposes
- Moerdijk, Vermeulen
(Show Context)
Citation Context ... theory of cell-like maps in the language of ∞-topoi. 7.3.1 Proper Maps of ∞-Topoi In this section, we introduce the notion of a proper geometric morphism between ∞-topoi. Here we follow the ideas of =-=[39]-=-, and turn the conclusion of the proper base change theorem into a definition. First, we require a bit of terminology. Suppose given a diagram of categories and functors � Y p C ′ q ′ ∗ �� D ′ C p ′ ∗... |

30 | Parametrized Homotopy Theory - May, Sigurdsson - 2004 |

29 | Homotopical algebraic geometry I: topos theory, preprint available at math.AG/0207028 - Toën, Vezzosi |

27 |
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Citation Context ...ap H BM (X) → lim H ←− BM (U•) is not an equivalence. Let X be the Hilbert cube Q = [0, 1] × [0, 1] × . . . (more generally, we could take X to be any nonempty Hilbert cube manifold). It is proven in =-=[10]-=- that every point of X has arbitrarily small neighborhoods which are homeomorphic to Q × [0, 1). Consequently, there exists a hypercovering U• of X, where each Un is a disjoint union of open subsets o... |

24 | Homotopy Coherent Category Theory - Cordier, Porter - 1997 |

22 |
Shape Theory
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(Show Context)
Citation Context ...nderstood by restricting to points. Let us consider an example from geometric topology. A map f : X → Y of compact metric spaces is called cell-like if each fiber Xy = X ×Y {y} has trivial shape (see =-=[12]-=-). This notion has good formal properties provided that we restrict our attention to metric spaces which are absolute neighborhood retracts. In the general case, the theory of cell-like maps can be ba... |

21 |
Homotopy associativity of H-spaces I,II
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(Show Context)
Citation Context ...ative multiplication operation. If U• is effective, then there exists a fiber diagram U1 �� ∗ ∗ � U−1 so that U1 is homotopy equivalent to a loop space. This is an classical result (see, for example, =-=[47]-=-). We will give a somewhat indirect proof in the next section. 6.1.3 ∞-Topoi and Descent In this section, we will describe an elegant characterization of the notion of an ∞-topos, based on the theory ... |

21 | Vers une axiomatisation de la théorie des catégories supériures
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(Show Context)
Citation Context ... us to easily compare the theory of ∞-categories with other models of higher category theory, such as simplicial categories. There is another approach to obtaining comparison results, due to Toën. In =-=[50]-=-, he shows that if C is a model category equipped with a cosimplicial object C • satisfying certain conditions, then C is (canonically) Quillen equivalent to Rezk’s category of complete Segal spaces. ... |

20 | Homotopy Theory of Simplicial Groupoids - Dwyer, Kan - 1984 |

19 | Strong stacks and classifying spaces - Joyal, Tierney - 1991 |

19 | On Homotopy Varieties - Rosicky |

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

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(Show Context)
Citation Context ...r model categories but it is more complicated: any suitably nice model category is equivalent to a model category enriched over the category Set∆ of simplicial sets, in an essentially unique way: see =-=[43]-=- for a precise statement and a proof. A.2.8 Enriched Quillen Adjunctions In this section, we assume that S is a monoidal model category in which every object is cofibrant. Let C and D be model categor... |

16 | A survey of (∞, 1)-categories. Available at math.AT/0610239 - Bergner |

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14 |
Homotopy Invariant Structures on Topological Spaces
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(Show Context)
Citation Context ...but difficult to work with when one wishes to perform even simple categorical constructions. As a remedy, we will introduce the more convenient formalism of ∞-categories (called weak Kan complexes in =-=[7]-=- and quasi-categories in [30]), which provides a more suitable setting for adaptations of sophisticated category-theoretic ideas. Our goal in §1.1.1 is to introduce both approaches and to explain why ... |