# A Survey of (∞, 1)-Categories

@inproceedings{Bergner2010ASO,
title={A Survey of (∞, 1)-Categories},
author={Julia E. Bergner},
year={2010}
}
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.
77 Citations
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#### References

SHOWING 1-10 OF 59 REFERENCES
A remark on K-theory and S-categories
• Mathematics
• 2002
Abstract It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is toExpand
Universal Homotopy Theories
Abstract Begin with a small category C . The goal of this short note is to point out that there is such a thing as a “universal model category built from C .” We describe applications of this to theExpand
A model for the homotopy theory of homotopy theory
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or moreExpand
Quasi-categories and Kan complexes
A quasi-category X is a simplicial set satisfying the restricted Kan conditions of Boardman and Vogt. It has an associated homotopy category hoX. We show that X is a Kan complex iff hoX is aExpand
Function complexes in homotopical algebra
• Mathematics
• 1980
1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4Expand
HOMOTOPY COMMUTATIVE DIAGRAMS AND THEIR REALIZATIONS
• Mathematics
• 1989
Abstract In this paper we describe an obstruction theory for the problem of taking a commutative diagram in the homotopy category of topological spaces and lifting it to an actual commutative diagramExpand
The homotopy theory of dg-categories and derived Morita theory
The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D inExpand
CHAPTER 2 – Homotopy Theories and Model Categories
• Mathematics
• 1995
This chapter explains homotopy theories and model categories. A model category is just an ordinary category with three specified classes of morphisms—fibrations, cofibrations, and weakExpand
A CHARACTERIZATION OF FIBRANT SEGAL CATEGORIES
In this note we prove that Reedy fibrant Segal categories are fi- brant objects in the model category structure SeCatc. Combining this result with a previous one, we thus have that the fibrantExpand
A model category structure on the category of simplicial categories
In this paper we put a cofibrantly generated model category struc- ture on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence ofExpand