# Algebraic models for homotopy types

## Frequency

Only this year

## Summary

ln this course we will develop algebraic and coalgebraic models for homotopy types. Among other things we will learn about Quillen's and Sullivan's model of rationâl homotopy types and about Mandell's theorem in p-adic homotopy theory.

## Content

We will start with an introduction to rational homotopy theory and prove Quillen's and Sullivan's theorem providing a Lie model, coalgebra model and algebra model for rational homotopy types. Moreover we will learn about Koszul duality relating the Lie model to the coalgebra model.

We will continue with p-adic homotopy theory and prove Mandell's theorem classifying p-adic homotopy types by E_infinity-algebras.

To prove Mandell's theorem we will learn about E_infinity-algebras and unstable algebras.

Besides that we will learn about the higher algebra and higher category theory used to prove these results.

## Keywords

Rational homotopy theory, p-adic homotopy theory, Sullivan model, Koszul duality, E_infinity-algebras, unstable algebras

## Learning Prerequisites

## Required courses

Some familiarity with homotopy theory, category theory.

## Learning Outcomes

- why algebraic models for homotopy types are useful.
- Translate between different models of homotopy types.

## Resources

## Bibliography

Classical sources for rational homotopy theory

Mandell's article: E_infinity algebras and p-adic homotopy theory

## Ressources en bibliothèque

## In the programs

**Exam form:**Project report (session free)**Subject examined:**Algebraic models for homotopy types**Lecture:**28 Hour(s)