## A SURVEY OF (∞, 1)-CATEGORIES (2006)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Bergner06asurvey,

author = {Julia E. Bergner},

title = {A SURVEY OF (∞, 1)-CATEGORIES},

year = {2006}

}

### OpenURL

### Abstract

Abstract. In this paper we give a summary of the comparisons between different definitions of so-called (∞,1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories. 1.

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Citation Context ... and Spalinski mention this concept at the end of their survey paper [13], and the idea was further explored by Rezk [27], whose ideas we will return to in the next section. The author then showed in =-=[2]-=- that the category of all small simplicial categories with the DK-equivalences has a model category structure, thus formalizing the idea. In order to define the fibrations in this model structure, we ... |

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Citation Context ...hat of a Segal ncategory and give a model structure for Segal n-categories for any n ≥ 1. The idea behind this generalization is used for both the Simpson and Tamsamani definitions of weak n-category =-=[30]-=-, [33]. The author gives a new proof of this model structure, just for the case of Segal categories, from which it is easier to characterize the fibrant objects [3]. It should be noted that, as in the... |

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Citation Context ...egories. From the viewpoint of higher category theory, these comparisons provide a kind of baby version of the comparisons which are being attempted between various definitions of weak n-category. In =-=[37]-=-, Toën actually axiomatizes a theory of (∞, 1)categories and proves that any such theory is equivalent to the theory of complete Segal spaces. In [36], he sketches arguments for proving the equivalenc... |

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Citation Context ...X and the simplicial resolution C∗[n] of the category [n] = (0 → · · · → n), the coherent nerve Ñ(X) is defined by Ñ(X)n = HomSC(C∗[n], X). This functor has a left adjoint J : QCat → SC. Theorem 7.8. =-=[19]-=- The adjoint pair is a Quillen equivalence. J : QCat �� SC : Ñ��� �� �� �� ���� ���� ��� �� ���� 12 J.E. BERGNER Thus, we have the following diagram of Quillen equivalences of model categories: SC � ... |

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Citation Context ...eories. A complete Segal space is first a simplicial space. It should be noted that we require that certain of our objects be fibrant in the Reedy model structure on the category of simplicial spaces =-=[26]-=-. This structure is defined by levelwise weak equivalences and cofibrations, but its importance here is that several of our constructions will be homotopy invariant because the objects involved satisf... |

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Citation Context ...etween various definitions of weak n-category. In [37], Toën actually axiomatizes a theory of (∞, 1)categories and proves that any such theory is equivalent to the theory of complete Segal spaces. In =-=[36]-=-, he sketches arguments for proving the equivalences between the four structures used in this paper. Although some of the functors he suggests do not appear to give the desired Quillen equivalences (o... |

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Citation Context ...55-02. 12 J.E. BERGNER It should be noted that there are other proposed models, including that of A∞categories. Joyal and Tierney briefly discuss some other approaches to this idea in their epilogue =-=[21]-=-. Furthermore, we should also mention that these structures are of interest in areas beyond homotopy theory and higher category theory. For instance, there are situations in algebraic geometry in whic... |

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Citation Context ...in this paper, but from the point of view of homotopy theory, they might be the easiest to use because the corresponding model structure gives what Dugger calls a presentation for the homotopy theory =-=[7]-=-. They are defined by Rezk [27] whose purpose was explicitly to find a nice model for the homotopy theory of homotopy theories. A complete Segal space is first a simplicial space. It should be noted t... |

14 |
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Citation Context ...derived category D(X) does not seem to determine its Ktheory spectrum, whereas its simplicial localization L(X) does [38]. Furthermore, the simplicial category L(X) forms a stack, which D(X) does not =-=[16]-=-. Similar work is also being done using dg categories, which are in many ways analogous to simplicial categories [35]. In particular, the category of dg categories has a model category structure which... |

9 |
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Citation Context ... paper, but it seems likely that they should hold. Some of the comparisons are also mentioned in work by Simpson, who sketches an argument for comparing the Segal categories and complete Segal spaces =-=[31]-=-. The author showed in [3] that simplicial categories are equivalent to Segal categories, which are in turn equivalent to the complete Segal spaces. However, the adjoint pairs go in opposite direction... |

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Citation Context ... a Segal ncategory and give a model structure for Segal n-categories for any n ≥ 1. The idea behind this generalization is used for both the Simpson and Tamsamani definitions of weak n-category [30], =-=[33]-=-. The author gives a new proof of this model structure, just for the case of Segal categories, from which it is easier to characterize the fibrant objects [3]. It should be noted that, as in the case ... |

6 |
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Citation Context ...the more commonly used derived category cannot. Given a scheme X, for example, its derived category D(X) does not seem to determine its Ktheory spectrum, whereas its simplicial localization L(X) does =-=[38]-=-. Furthermore, the simplicial category L(X) forms a stack, which D(X) does not [16]. Similar work is also being done using dg categories, which are in many ways analogous to simplicial categories [35]... |

5 |
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Citation Context ...ive a Quillen equivalence, partly because their work predates both model structures by several years. Schwänzl and Vogt also address this question, using topological rather than simplicial categories =-=[28]-=-. Rezk defines complete Segal spaces with the comparison with simplicial categories in mind [27]. While his functor from simplicial categories to complete Segal spaces naturally factors through Segal ... |

4 |
Higher topos theory, preprint available at math.CT/0608040
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Citation Context ...n the category of simplicial categories [32]. Also motivated by ideas in algebraic geometry, Lurie uses quasi-categories and their relationship with simplicial categories in his work on higher stacks =-=[22]-=-. The first chapter of his manuscript is also a good introduction to many of the ideas of (∞, 1)-categories. Another application of the model category of simplicial categories can be found in recent w... |

3 | A characterization of fibrant Segal categories
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(Show Context)
Citation Context ...ining Segal spaces, for which we need the Segal map assigned to a simplicial space. As one might guess from its name, the Segal map is first defined by Segal in his work with Γ-spaces [29]. Let α i : =-=[1]-=- → [k] be the map in ∆ such that α i (0) = i and α i (1) = i+1, defined for each 0 ≤ i ≤ k −1. We can then define the dual maps αi : [k] → [1] in ∆ op . For k ≥ 2, the Segal map is defined to be the m... |

1 |
Twisted parametrized stable homotopy theory, preprint available at math.AT/0508070
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Citation Context ...ion to many of the ideas of (∞, 1)-categories. Another application of the model category of simplicial categories can be found in recent work of Douglas on twisted parametrized stable homotopy theory =-=[6]-=-. He uses diagrams of simplicial categories weakly equivalent to the simplicial localization of the category of spectra (i.e., equivalent as homotopy theories to the homotopy theory of spectra) in ord... |

1 |
Segal categories and quasicategories, preprint available at math.AT/0401274
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(Show Context)
Citation Context ... desired Quillen equivalences (or at any rate are not used in the known proofs), he gives a good overview of the problem. Another good introduction of the problem can be found in a preprint by Porter =-=[24]-=-, and a nice description of the idea behind (∞, 1)-categories can be found in [34]. From another point of view, these comparisons are of interest in homotopy theory, as what we are here calling a (∞, ... |