Workshop on the homotopy theory of homotopy theories

These notes are from a series of lectures given at the Workshop on the Homotopy Theory of Homotopy Theories which took place in Caesarea, Israel, in May 2010. The workshop was organized by David Blanc, Emmanuel Farjoun, and David Kazhdan, and talks not indicated otherwise were given by the author.

💡 Research Summary

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These notes record the series of lectures delivered at the “Workshop on the Homotopy Theory of Homotopy Theories” held in Caesarea, Israel, in May 2010. The workshop, organized by David Blanc, Emmanuel Farjoun and David Kazhdan, was largely presented by Julia E. Bergner. The central theme is the meta‑question: once one has a homotopy theory (a category equipped with a distinguished class of weak equivalences), how can one treat the collection of all such homotopy theories as a homotopy theory itself?

The exposition begins with the classical viewpoint: the homotopy theory of topological spaces is obtained by localizing the category of spaces at the weak homotopy equivalences. Quillen’s model‑category axioms give a systematic way to replace an arbitrary category with weak equivalences by a richer structure (cofibrations, fibrations, and weak equivalences) that permits homotopical constructions such as homotopy limits, colimits, and mapping spaces. The author reviews the basic examples—localization of groups to abelian groups, the derived category of chain complexes, and the two standard model structures on Top (Strøm’s model and the Quillen–Hurewicz model).

Next, the Dwyer–Kan simplicial localization is introduced. Given a category C and a subcategory S of weak equivalences (with the same objects), one can construct a simplicial category L(C,S) whose hom‑objects are simplicial sets encoding derived mapping spaces. The key theorem (Dwyer–Kan) states that π₀ L(C,S) recovers the ordinary localization C