Model Categories

Winter 2023

In this quarter, we will be learning about model categories. Historically, model categories were introduced by Quillen as a setting where one can do homotopy theory. Quillen used the formalism of model categories in his work on rational homotopy theory, proving the homotopy theory of simply connected spaces under a coarse notion of equivalence is the same as studying rational commutative differential graded algebras. More  recently, they have been used in Morel and Voevodsky’s work in A1-homotopy theory, which gives a way to create a homotopy category in the context of schemes. We will study the theory and properties of Quillen model categories with an emphasis on examples in familiar categories.

We will primarily be following Scott Balchin's recent book A Handbook of Model Categories. Particular sections of emphasis will be 2.1-2.4, 3.1-3.3, 4.1, 6.1, 6.2, 7.1, 8.1, 8.2. Notably, this text truly is a handbook. It contains very little, if any, proofs. The proofs will be supplemented via Mark Hovey's book Model Categories. As an additional reference, it should be noted that Quillen's original treatment in Homotopical Algebra, though terse, has recently been typed up here.

The seminar will be discussion and presentation based. The meetings will be every Friday from 12-1 in THO 231. 

1/6  Introduction, history of model categories, assignment of presentations

1/13  Sections 2.1 and 2.2, Model categories and the homotopy category (Presenter: Jackson Morris)

1/20  Sections 2.3 and 2.4, Quillen adjunctions and equivalences and the small object argument (Presenter: Raymond Guo)

1/27 Sections 3.1, 3.2, and 3.3, Cofibrantly generated, combinatorial, and cellular model categories (Presenter: Soham Ghosh) 

2/3  Section 4.1,  Left Bousfield localization (Presenter: Justin Bloom)

2/10 Simplicial sets (Presenter: Nelson Niu)

2/17 More on Bousfield localization (Presenter: Alex Waugh) [Reference: https://arxiv.org/abs/2002.03888]