# Homotopy Theory of Diagrams

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*Wojciech Chachólski; Jérôme Scherer*

In this paper we develop homotopy theoretical methods for
studying diagrams. In particular we explain how to construct homotopy colimits
and limits in an arbitrary model category. The key concept we introduce is
that of a model approximation. A model approximation of a category
\(\mathcal{C}\) with a given class of weak equivalences is a model
category \(\mathcal{M}\) together with a pair of adjoint functors
\(\mathcal{M} \rightleftarrows \mathcal{C}\) which satisfy certain
properties. Our key result says that if \(\mathcal{C}\) admits a model
approximation then so does the functor category \(Fun(I,
\mathcal{C})\).

From the homotopy theoretical point of view categories with
model approximations have similar properties to those of model categories. They
admit homotopy categories (localizations with respect to weak equivalences).
They also can be used to construct derived functors by taking the analogs of
fibrant and cofibrant replacements.

A category with weak equivalences can have several useful
model approximations. We take advantage of this possibility and in each
situation choose one that suits our needs. In this way we prove all the
fundamental properties of the homotopy colimit and limit: Fubini Theorem (the
homotopy colimit -respectively limit- commutes with itself), Thomason's theorem
about diagrams indexed by Grothendieck constructions, and cofinality
statements. Since the model approximations we present here consist of certain
functors “indexed by spaces”, the key role in all our arguments is
played by the geometric nature of the indexing categories.

#### Table of Contents

# Table of Contents

## Homotopy Theory of Diagrams

- Contents vii8 free
- Introduction 112 free
- Chapter I. Model approximations and bounded diagrams 516 free
- 1. Notation 516
- 2. Model categories 617
- 3. Left derived functors 1223
- 4. Left derived functors of colimits and left Kan extensions 1425
- 5. Model approximations 1627
- 6. Spaces and small categories 1930
- 7. The pull-back process and local properties 2233
- 8. Colimits of diagrams indexed by spaces 2233
- 9. Left Kan extensions 2435
- 10. Bounded diagrams 2637

- Chapter II. Homotopy theory of diagrams 2940
- 11. Statements of the main results 2940
- 12. Cofibrations 3041
- 13. Fun[sup(b)](K,M) as a model category 3243
- 14. Ocolimit of bounded diagrams 3546
- 15. Bousfield-Kan approximation of Fun(I,C) 3647
- 16. Homotopy colimits and homotopy left Kan extensions 3748
- 17. Relative boundedness 3849
- 18. Reduction process 4051
- 19. Relative cofibrations 4455
- 20. Cofibrations and colimits 4657
- 21. Fun[sup[(b)][sub(f)](L,M) as a model category 4859
- 22. Cones 4960
- 23. Diagrams indexed by cones I 5162

- Chapter III. Properties of homotopy colimits 5465
- Appendix A. Left Kan extensions preserve boundedness 7586
- Appendix B. Categorical Preliminaries 8293
- 34. Categories over and under an object 8293
- 35. Relative version of categories over and under an object 8293
- 36. Pull-back process and Kan extensions 8394
- 37. Cofinality for colimits 8394
- 38. Grothendieck construction 8495
- 39. Grothendieck construction & the pull-back process 8495
- 40. Functors indexed by Grothendieck constructions 8596

- Bibliography 8798
- Index 89100