# THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES

@article{Bergner2005THREEMF,
title={THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES},
author={Julia E. Bergner},
journal={Topology},
year={2005},
volume={46},
pages={397-436}
}
Abstract Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an… Expand
126 Citations

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#### References

SHOWING 1-10 OF 34 REFERENCES
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
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
Algebraic theories in homotopy theory
It is well known in homotopy theory that given a loop space X one can always find a simplicial group G weakly equivalent to X, such that the weak equivalence can be realized by maps preservingExpand
Rigidification of algebras over multi-sorted theories
We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result forExpand
Homotopical algebraic geometry. I. Topos theory.
• Mathematics
• 2002
This is the rst of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this rst part we investigate a notion of higher topos.Expand
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
Simplicial monoids and Segal categories
Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. In particular, we show that the usual model categoryExpand
Model categories and their localizations
Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left BousfieldExpand
Handbook of algebraic topology
Foreword. List of Contributors. Homotopy types (H.-J. Baues). Homotopy theories and model categories (W.G. Dwyer, J. Spalinski). Proper homotopy theory (T. Porter). Introduction to fibrewise homotopyExpand
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