**Cours spécialisé ** (TA)

# Algèbre supérieure

### Yonatan Harpaz

** Contact :** `harpaz à math.univ-paris13.fr`

** Notes de cours :** https://www.math.univ-paris13.fr/~harpaz/lecture_notes.pdf

## Présentation

This is an advanced course in homotopy theory intended for students with background in the feld. The course will focus on the notion of little $n$-cube algebras, a homotopy coherent algebraic structure interpolating between associative ($n=1$) and commutative ($n=\infty$) algebras. Such structures appear in a variety of higher categorical contexts, from the study of $n$-fold loop spaces, through deformation theory and up to topological field theories. We will begin by reviewing the basic theory of $\infty$-categories and continue from there to $\infty$-operads and their algebras, focusing on the machinery that will be needed later on, such as tensor product of $\infty$-operads. In the second half of the course we will focus on the little $n$-cube operads and their properties, including Dunn's additivity theorem and May's recognition principle. In the final part of the course we will discuss factorization homology, a fundamental invariant of little $n$-cube algebras, which is a vast generalization of Hochschild homology for associative algebras. Our goal is to build up towards Poincaré-Koszul duality, following the approach of Lurie and Ayala-Francis.

## Contenu

- Recollections on $\infty$-categories
- Higher categorical algebraic structures and $\infty$-operads
- The little $n$-cube operads and their algebras
- Dunn's additivity theorem
- May's recognition principle
- Factorization homology

## Prérequis

A level of familiarity with the basics of higher category theory such as acquired by taking the course "Introduction à la théorie de l'homotopie" given by Najib Idrissi in the third trimester.
## Bibliographie

- Jacob Lurie. Higher Topos Theory.
* selected parts of chapter 2 *
- Jacob Lurie. Higher Algebra.
* selected parts of chapters 2 and 5 *