Introduction to Homotopy Theory (2019–2020)

M2 Fundamental Mathematics (S2) • Lectures • 24h • Last updated on

The goal of this course is to give an introduction to modern homotopy theory, its tools, and its applications, as well as to introduce the notion of \(\infty\)-category. We will essentially follow two examples: the founding example of topological spaces and the example of chain complexes (in the sense of homological algebra and algebraic topology). We will present the modern axiomatic treatement of homotopy theory – Quillen’s model categories – and we will explain the equivalence between topological spaces and simplicial sets. We will illustrate these methods through the example of rational homotopy theory, showing how the multiplicative structure of cochains – singular or de Rham – encode topological spaces up to rational homotopy.

Prerequisites. It is advised to have already taken a class on algebraic topology as well as an introduction to homological algebra.

Plan

  1. Model Categories.
  2. Quillen functors and derived functors.
  3. Comparison between simplicial sets and topological spaces.
  4. Rational homotopy.
  5. Notion of \(\infty\)-category theory.

Organization

Lectures will being in January 2020 and end in February 2020.

Bibliography

Works on homotopy theory:

Reminders on algebraic topology and homological algebra:

(The password is homotopie.)