nLab Homotopy Limit Functors on Model Categories and Homotopical Categories

This entry is about the book

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith,

  Homotopy Limit Functors on

  Model Categories and Homotopical Categories

  Mathematical Surveys and Monographs 113

  AMS 2004

  ISBN: 978-1-4704-1340-8

  pdf

on the theory of homotopical categories and model categories – a presentation of (∞,1)-categories –, their simplicial localization/homotopy categories and derived functors such as homotopy limit functors.

A historically important early manuscript draft of this book is

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Model Categories and More General Abstract Homotopy Theory, PDF

The draft serves as an original reference for the Kan recognition theorem (in §II.8) and the Kan transfer theorem (in §II.9), as well as cofibrantly generated model categories (in Chapter II). This material did not make it to the book.

Furthermore, this draft originates the modern definition of a model category (in §I.1.2), modifying the original definition of closed model category by Quillen by replacing finite (co)limits with small (co)limits and requiring factorizations to be functorial.

References

Compare also:

• Bruno Kahn, Georges Maltsiniotis, Structures de Dérivabilité

category: reference