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Chapter 2900: A model for the homotopy theory of homotopy theory
Section 6.: Complete Segal spaces
Proof (05UK) (cite)

Proof. This is immediate from (6.4, (4)). ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Charles Rezk

Original source: arXiv:math/9811037v3

In this chapter

Chapter overview

1. Introduction
1.1. Natural examples
1.2. Classifying diagrams and classification diagrams
1.3. Applications
1.4. Other models
1.5. Organization of the paper
1.6. Acknowledgments
2. Simplicial spaces
2.1. Spaces
2.2. The simplicial indexing category
2.3. Simplicial spaces
2.4. Reedy model category
2.5. Compatibility with cartesian closure
2.6. Proper model categories
3. Nerve constructions and classification diagrams
3.1. The nerve of a category
3.3. The classification diagram of a pair of categories
3.4. Discrete nerve construction
3.5. The classifying diagram of a category
3.10. Classification diagrams of functor categories
4. Segal spaces
4.1. Definition of a Segal space
4.4. Examples
5. Homotopy theory in a Segal space
5.1. “Objects” and “mapping spaces”
5.3. “Homotopies” and “compositions” of “maps”
5.5. Homotopy category and homotopy equivalences
5.7. The space of homotopy equivalences
6. Complete Segal spaces
7. Closed model category structures
7.4. Dwyer-Kan equivalences
8. Complete Segal spaces from model categories
8.1. Universes
8.2. The classification space of a closed model category
8.6. Results about classification spaces
8.9. Categories of diagrams
9. Localization model category
10. Segal space model category structure
11. Equivalences in Segal spaces
11.2. Morphisms induced by compositions
11.7. Proof of (11.1)
12. Complete Segal space closed model category structure
12.2. Homotopy monomorphisms
12.3. Proof of (12.1)
13. Categorical equivalences
13.1. Categorical homotopies
13.3. Categorical equivalences
13.7. Categorical equivalences are Dwyer-Kan equivalences
14. A completion functor
References

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Original source · math/9811037v3