Eugenio Landi

Back to main page

Model Categories and Abstract Homotopy Theory

Program

The course will be an introduction to the theory of model categories and abstract homotopy theory. After the basic definitions (model structures, homotopy categories, Quillen functors) we will study homotopy limits and colimits, some useful properties of model categories and their relation to (∞,1)-categories. Time permitting we will talk about operads and the homotopy transfer theorem.

Lecture Links Topics Covered Lecture Notes
Lecture 1 Part 1  |  Part 2
  • Localization of Categories
  • Definition of Model Categories
  • Ken Brown Lemma
 Lecture 1
Lecture 2 Part 1  |  Part 2
  • Cylinder and Path Objects and Homotopies
  • Homotopy Categories and related results
 Lecture 2
Lecture 3 Part 1  |  Part 2
  • The Quillen Model Structure on Top and SSet
  • Quillen Adjunctions and Equivalences
 Lecture 3
Lecture 4 Part 1  |  Part 2
  • Homotopy (Co)limits as Derived Functors
  • Model Structures on Categories of Functors
    • Projective and Injective Model Structures
    • Reedy Categories and Reedy Model Structures
 Lecture 4
Lecture 5 Part 1  |  Part 2
  • Homotopy Pushouts and Pullbacks as Pushouts and Pullbacks
  • (∞,1)-Categories as SSet-enriched Categories
  • Dwyer-Kan Simplicial Localization of a Category with Weak Equivalences
  • Simplicial Localization of Model Categories via Framings following Hovey
 Lecture 5