## Quasi-categories vs Segal spaces (2006)

Venue: | IN CATEGORIES IN ALGEBRA, GEOMETRY AND MATHEMATICAL |

Citations: | 12 - 0 self |

### BibTeX

@INPROCEEDINGS{Joyal06quasi-categoriesvs,

author = {André Joyal and Myles Tierney},

title = { Quasi-categories vs Segal spaces},

booktitle = {IN CATEGORIES IN ALGEBRA, GEOMETRY AND MATHEMATICAL},

year = {2006},

pages = {277--326},

publisher = {}

}

### OpenURL

### Abstract

We show that complete Segal spaces and Segal categories are Quillen equivalent to quasi-categories.

### Citations

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Citation Context ...esian closed model category, if f is a fibration and u is a cofibration, then the map 〈u, f〉 is a fibration, which is acyclic if u or f is acyclic. We recall a few notions of enriched category theory =-=[K]-=-. Let V = (V, ⊗, σ) a bicomplete symmetric monoidal closed category. A category enriched over V is called a V-category. If A and B are V-categories, there is a notion of a strong functor F : A → B; it... |

364 |
Homotopical algebra
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Citation Context ...perty with respect to every horn inclusion Λ k [n] ⊂ ∆[n]. The following theorem describes the classical model structure on the category S. f4 ANDRÉ JOYAL AND MYLES TIERNEY Theorem 1.1 (Quillen, see =-=[Q]-=-). The category of simplicial sets S admits a model structure (C0, W0, F0) in which a cofibration is a monomorphism, a weak equivalence is a weak homotopy equivalence and a fibration is a Kan fibratio... |

198 |
Vogt – Homotopy invariant algebraic structures on topological spaces
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Citation Context ...gal spaces and Segal categories are Quillen equivalent to quasi-categories. Introduction Quasi-categories were introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures =-=[BV]-=-. They are often called weak Kan complexes in the literature. The category of simplicial sets S admits a Quillen model structure in which the fibrant objects are the quasi-categories by a result of th... |

183 | Simplicial homotopy theory - Goerss, Jardine - 1999 |

134 | Homotopy theories and model categories, Handbook of algebraic topology - Dwyer, Spalinski - 1995 |

104 |
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Citation Context ...een simplicial sets is usually defined by using the geometric realisation functor S → Top. An alternative definition uses Kan complexes and the homotopy category S π0 introduced by Gabriel and Zisman =-=[GZ]-=-. Recall that a simplicial set X is called a Kan complex if every horn Λ k [n] → X has a filler ∆[n] → X. If A, B ∈ S let us put � X � �� Y π0(A, B) = π0(B A ). If we apply the functor π0 to the compo... |

100 | On closed categories of functors - Day - 1974 |

59 |
Function complexes in homotopical algebra, Topology 19
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Citation Context ...rmits the construction of homotopy pullbacks and pushouts, of fiber and cofiber sequences, etc. Dwyer and Kan have proposed using general simplicial categories to model general homotopy theories, see =-=[DK1]-=- and [DK2]. Recently, Bergner [B1] has established a model structure on the category of simplicial categories. In [B2], she introduces a new model structure Segal cat ′ on the category of Segal precat... |

52 | Higher Topos Theory
- Lurie
- 2009
(Show Context)
Citation Context ...rst author in [J2]. We call this model structure, the model structure for quasicategories. The theory of quasi-categories has applications to homotopical algebra and higher category theory, see [J3], =-=[Lu1]-=- and [Lu2]. Complete Segal spaces were introduced by Rezk in his work on the homotopy theory of homotopy theories [Rez]. The category of bisimplicial sets S (2) admits a Quillen model structure in whi... |

48 | A model category structure on the category of simplicial categories
- Bergner
(Show Context)
Citation Context ...pullbacks and pushouts, of fiber and cofiber sequences, etc. Dwyer and Kan have proposed using general simplicial categories to model general homotopy theories, see [DK1] and [DK2]. Recently, Bergner =-=[B1]-=- has established a model structure on the category of simplicial categories. In [B2], she introduces a new model structure Segal cat ′ on the category of Segal precategories and she obtains a chain of... |

46 | Three models for the homotopy theory of homotopy theories
- Bergner
(Show Context)
Citation Context ...category of precategories admits a model structure in which the fibrant objects are Segal categories. We call this model structure the model structure for Segal categories. By a theorem of Bergner in =-=[B2]-=-, the The research of the first author was supported by the NSERC. 12 ANDRÉ JOYAL AND MYLES TIERNEY inclusion functor π∗ : PCat ⊂ S (2) induces a Quillen equivalence between the model structure for S... |

45 |
Notes on quasi-categories
- Joyal
(Show Context)
Citation Context ...alled weak Kan complexes in the literature. The category of simplicial sets S admits a Quillen model structure in which the fibrant objects are the quasi-categories by a result of the first author in =-=[J2]-=-. We call this model structure, the model structure for quasicategories. The theory of quasi-categories has applications to homotopical algebra and higher category theory, see [J3], [Lu1] and [Lu2]. C... |

40 |
Homotopy commutative diagrams and their realizations
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- 1989
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Citation Context ... in the opposite direction, t! : S (2) ↔ S : t ! , where the functor t! associates to a bisimplicial set X a total simplicial set t!(X). Segal categories were first introduced by Dwyer, Kan and Smith =-=[DKS]-=-, where they are called special ∆ o -diagrams of simplicial sets . The theory of Segal categories was extensively developed by Hirschowitz and Simpson for application to algebraic geometry. They show ... |

40 | A model for the homotopy theory of homotopy theory
- Rezk
(Show Context)
Citation Context ...es has applications to homotopical algebra and higher category theory, see [J3], [Lu1] and [Lu2]. Complete Segal spaces were introduced by Rezk in his work on the homotopy theory of homotopy theories =-=[Rez]-=-. The category of bisimplicial sets S (2) admits a Quillen model structure in which the fibrant objects are the complete Segal spaces. We call this model structure the model structure for complete Seg... |

35 | Quasi-categories and Kan complexes - Joyal - 2002 |

34 |
Model Categories and Their
- Hirschhorn
- 2003
(Show Context)
Citation Context ...fibrant objects to a weak equivalence. Corollary 7.11. A left Quillen functor takes a weak equivalence between cofibrant objects to a weak equivalence. The following result is due to Reedy [Ree]. See =-=[Hi]-=- and [JT2]. Proposition 7.12. The cobase change along a cofibration of a weak equivalence between cofibrant objects is a weak equivalence. Corollary 7.13. If every object of model category is cofibran... |

22 |
Homotopy coherent category theory and A∞-structures in monoidal categories
- Batanin
- 1998
(Show Context)
Citation Context ...proving all the equivalences above was proposed by Töen in [T2]. There are other important notions of homotopy theories, for example the A∞spaces of Stasheff [MSS]. See also Batanin for A∞-categories =-=[Ba]-=-. A theory of homotopical categories was developed by Dwyer, Hirschhorn, Kan and Smith [DHKS]. Dugger studies universal homotopy theories in [D]. The model structure for quasi-categories belongs to a ... |

22 |
Vogt’s Theorem on Categories of Homotopy Coherent Diagrams
- Cordier, Porter
- 1986
(Show Context)
Citation Context ...llen48 ANDRÉ JOYAL AND MYLES TIERNEY equivalent to the model structure for quasi-categories. We shall see in [J4] that the coherent nerve functor defined by Cordier [C] and studied in Cordier-Porter =-=[CP]-=- defines a direct Quillen equivalence Simp.cat → Quasicat between the model structure for simplicial categories and the model structure for quasi-categories. See Lurie [Lu1] for another proof. An axio... |

22 |
Homotopy theories
- Heller
- 1988
(Show Context)
Citation Context ... [S]. A theory of homotopical categories was developed by Dwyer, Kan, Hirschhorn and Smith [DHKS]. Dugger studies universal homotopy theories in [D]. A quite different notion was introduced by Heller =-=[He]-=- based on the idea of hyperdoctrine of Lawvere [La]. A similar notion called dérivateur was later introduced by Grothendieck and studied by Maltsiniotis [M].48 ANDRÉ JOYAL AND MYLES TIERNEY Reference... |

21 | Vers une axiomatisation de la théorie des catégories supériures
- Toën
(Show Context)
Citation Context ...tructure for simplicial categories and the model structure for quasi-categories. See Lurie [Lu1] for another proof. An axiomatic approach to proving all the equivalences above was proposed by Töen in =-=[T2]-=-. There are other important notions of homotopy theories, for example the A∞spaces of Stasheff [MSS]. See also Batanin for A∞-categories [Ba]. A theory of homotopical categories was developed by Dwyer... |

19 |
Simplicial categories vs. quasi-categories
- Joyal
(Show Context)
Citation Context ...pson model structure or the model structure for Segal categories. The model structure is cartesian closed by a result of Pellisier in [P]. A precategory is fibrant iff it is a Segal space by [B3] and =-=[J4]-=-. A simplicial set X : ∆ o → Set is discrete iff it takes every map in ∆ to a bijection. It follows that a bisimplicial set X : (∆ 2 ) o → Set is a precategory iff it takes every map in [0] × ∆ to a b... |

19 |
Strong stacks and classifying spaces
- Joyal, Tierney
- 1991
(Show Context)
Citation Context ...e groupoid generated by one isomorphism 0 → 1. We say that a functor A → B is monic on objects (resp. surjective on objects) if the induced map Ob(A) → Ob(B) is monic (resp. surjective). Theorem 1.4. =-=[JT1]-=- [Rez] The category Cat admits a model structure in which a cofibration is a functor monic on objects, a weak equivalence is an equivalence of categories and a fibration is a quasi-fibration. The acyc... |

17 |
Homotopy theory of model categories, unpublished manuscript, available at http://www-math.mit.edu/~psh
- Reedy
(Show Context)
Citation Context ... is acyclic for every n ≥ 0. We shall say that f is a Reedy cofibration if it is a Reedy C-cofibration. We shall say that f is a Reedy fibration if it is a Reedy F-fibration. Theorem 7.35 (Reedy, see =-=[Ree]-=-). Let (C, W, F) be a model structure on a category E. Then the category [∆ o , E] admits a model structure (C ′ , W ′ , F ′ ) in which C ′ is the class of Reedy cofibrations, W ′ is the class of leve... |

17 | Simplicial structures on model categories and functors
- Rezk, Schwede, et al.
- 1999
(Show Context)
Citation Context ...implicial. However, it is Quillen equivalent to the model category for complete Segal spaces, which is simplicial by Theorem 4.1. In their paper Simplicial structures on model categories and functors =-=[RSS]-=- Rezk, Schwede and Shipley study the problem of associating to a model category E a Quillen equivalent simplicial model category. A simplicial object X : ∆ o → E is said to be homotopically constant i... |

17 |
Homotopical and higher categorical structures in algebraic geometry, Hablitation Thesis available at math.AG/0312262
- Toën
(Show Context)
Citation Context ... j ∗ : PCat → S is right adjoint to the functor q ∗ . If X is a precategory, then j ∗ (X) is the first row of X. If A ∈ S, then q ∗ (A) = A□1. The following result was conjectured by Bertrand Töen in =-=[T1]-=-: Theorem 5.6. The adjoint pair of functors q ∗ : S ↔ PCat : j ∗ is a Quillen equivalence between the model category for quasi-categories and the model category for Segal categories.QUASI-CATEGORIES ... |

14 |
Universal homotopy theories, Adv
- Dugger
(Show Context)
Citation Context ... of Stasheff [MSS]. See also Batanin for A∞-categories [Ba]. A theory of homotopical categories was developed by Dwyer, Hirschhorn, Kan and Smith [DHKS]. Dugger studies universal homotopy theories in =-=[D]-=-. The model structure for quasi-categories belongs to a class of model structures in presheaf categories studied by Cisinski [Ci]. A quite different notion of homotopy theory was introduced by Heller ... |

14 |
Descente pour les n-champs, preprint available at math.AG/9807049
- Hirschowitz, Simpson
(Show Context)
Citation Context ...u : [m] → [n] is a map in ∆, then the map X(u) : Xn → Xm induces a map for every a ∈ X [n]0 0 . X(a0, a1, . . . , an) → X(a u(0), a u(1), . . . , a u(m)) A precategory X is called a Segal category in =-=[HS]-=- if it satisfies the Segal condition 3.1. It is easy to verify that we have a decomposition In\X = ⊔ X(a0, a1) × · · · × X(an−1, an). a∈X [n] 0 0 It follows that a precategory X is a Segal category if... |

13 |
la notion de diagramme homotopiquement coherent", Third Colloquium on Categories
- Cordier, \Sur
- 1982
(Show Context)
Citation Context ...icial categories is indirectly Quillen48 ANDRÉ JOYAL AND MYLES TIERNEY equivalent to the model structure for quasi-categories. We shall see in [J4] that the coherent nerve functor defined by Cordier =-=[C]-=- and studied in Cordier-Porter [CP] defines a direct Quillen equivalence Simp.cat → Quasicat between the model structure for simplicial categories and the model structure for quasi-categories. See Lur... |

11 | Introduction à la théorie des dérivateurs (d’après Grothendieck). http: //people.math.jussieu.fr/~maltsin/textes.html - Maltsiniotis - 2001 |

9 |
Closed categories, lax limits, and homotopy limits
- Gray
- 1980
(Show Context)
Citation Context ...es ⊙ : E1 × E2 → E3 is divisible on both sides, then so are the left division functor E o 1 × E3 → E2 and the right division functor E3 × E o 2 → E1. This is called a tensor-hom-cotensor situation in =-=[G]-=-. Recall that a monoidal category E = (E, ⊗) is said to be closed if the tensor product ⊗ is divisible on each side. Let E = (E, ⊗, σ) be a symmetric monoidal�� �� �� � � � �� �� �� QUASI-CATEGORIES ... |

9 |
Catégories enrichies faibles, Thèse, Université de Nice-Sophia Antipolis
- Pellissier
- 2002
(Show Context)
Citation Context ...cartesian closed. We call the model structure, the Hirschowitz-Simpson model structure or the model structure for Segal categories. The model structure is cartesian closed by a result of Pellisier in =-=[P]-=-. A precategory is fibrant iff it is a Segal space by [B3] and [J4]. A simplicial set X : ∆ o → Set is discrete iff it takes every map in ∆ to a bijection. It follows that a bisimplicial set X : (∆ 2 ... |

5 |
A survey of
- Bergner
(Show Context)
Citation Context ...categories and she obtains a chain of Quillen equivalences: Simp.cat → Segal cat ′ ← Segal cat ← Comp. Segal sp (in this chain, a Quillen equivalence is represented by the right adjoint functor). See =-=[B4]-=- for a survey. It follows from her results combined the result proved in the present paper that the model structure for simplicial categories is indirectly Quillen48 ANDRÉ JOYAL AND MYLES TIERNEY equ... |

4 | Structures d’asphéricité, foncteurs lisses, et fibrations - Maltsiniotis - 2005 |

3 |
Simplicial localisations of categories
- Dwyer, Kan
- 1980
(Show Context)
Citation Context ...construction of homotopy pullbacks and pushouts, of fiber and cofiber sequences, etc. Dwyer and Kan have proposed using general simplicial categories to model general homotopy theories, see [DK1] and =-=[DK2]-=-. Recently, Bergner [B1] has established a model structure on the category of simplicial categories. In [B2], she introduces a new model structure Segal cat ′ on the category of Segal precategories an... |

2 |
Stable Infinity Categories
- Lurie
(Show Context)
Citation Context ... in [J2]. We call this model structure, the model structure for quasicategories. The theory of quasi-categories has applications to homotopical algebra and higher category theory, see [J3], [Lu1] and =-=[Lu2]-=-. Complete Segal spaces were introduced by Rezk in his work on the homotopy theory of homotopy theories [Rez]. The category of bisimplicial sets S (2) admits a Quillen model structure in which the fib... |

1 |
A characterisation of fibrant Segal categories, preprint available at arXiv:math.AT/0603400
- Bergner
(Show Context)
Citation Context ...owitz-Simpson model structure or the model structure for Segal categories. The model structure is cartesian closed by a result of Pellisier in [P]. A precategory is fibrant iff it is a Segal space by =-=[B3]-=- and [J4]. A simplicial set X : ∆ o → Set is discrete iff it takes every map in ∆ to a bijection. It follows that a bisimplicial set X : (∆ 2 ) o → Set is a precategory iff it takes every map in [0] ×... |

1 |
Les préfaisceaux comme modèle des types d’ homotopie
- Cisinski
(Show Context)
Citation Context ...rn, Kan and Smith [DHKS]. Dugger studies universal homotopy theories in [D]. The model structure for quasi-categories belongs to a class of model structures in presheaf categories studied by Cisinski =-=[Ci]-=-. A quite different notion of homotopy theory was introduced by Heller [He] based on the idea of hyperdoctrine of Lawvere [La]. A similar notion called dérivateur was later introduced by Grothendieck ... |

1 |
Homotopy limit functors on Model
- Dwyer, Hirschhorn, et al.
- 2004
(Show Context)
Citation Context ...notions of homotopy theories, for example the A∞spaces of Stasheff [MSS]. See also Batanin for A∞-categories [Ba]. A theory of homotopical categories was developed by Dwyer, Hirschhorn, Kan and Smith =-=[DHKS]-=-. Dugger studies universal homotopy theories in [D]. The model structure for quasi-categories belongs to a class of model structures in presheaf categories studied by Cisinski [Ci]. A quite different ... |

1 |
Elements of Simplicial Homotopy Theory, in preparation
- Joyal, Tierney
(Show Context)
Citation Context ...alence is a weak homotopy equivalence and a fibration is a Kan fibration. The model structure is cartesian closed and proper. See Definition 7.7 in the appendix for the notion of model structure. See =-=[JT2]-=- for a purely combinatorial proof of the theorem. We call a map of simplicial sets a trivial fibration if it has the right lifting property with respect to every monomorphism. We note that this notion... |

1 | Lawvere, Equality in hyperdoctrines and comprehension scheme as an adjoint functor - W - 1970 |