Master II Course

Algebra + Homotopy = Operads

(November-December 2017)


``Algebra and homotopy theory do not mix well a priori'' used to say Saunders MacLane because algebraic structures are, in general, not stable under homotopy operations. On the other hand, Mathematics is full of algebraic structures on chain complexes for instance. In algebraic topology, the singular cochain complex of a topological space is endowed with a associative product. In differential geometry, the de Rham complex of a manifold carries a commutative product. And the deformation theory is all based on the notion of a differential graded Lie algebra.

One should consider MacLane's statement in a positive way: there is here a fruitful source of beautiful mathematical developments which would allow us to describe the homotopical properties of types of algebras, with a view toward applications in Algebra, Geometry, Topology, etc. The purpose of this course will be to get accustomed to some of the modern tools of homotopical algebra.  We will begin with a few but crucial examples of differential graded algebras and their properties, like the homotopy transfer theorem. We will then study how to encode conceptually the various types of algebraic structures with the notion of an operad. Operad theory, including the Koszul duality theory, will provide us with fundamental tools to describe the homotopical properties of differential graded algebras. We will conclude this course with recent and elegant applications to deformation theory (pre-Lie algebras and infiny-groupoid).

This course will directly follow the one of Muriel Livernet on Homological algebra and algebraic topology (September-October 2017); it will open the doors to the one of Gregory Ginot and Marco Robalo on an Introduction of homotopy theory (second semester, 2018).


  1. Differential graded algebras
  2. Operads
  3. Koszul duality
  4. Algebras up to homotopy
  5. Deformation theory



Les cours auront lieu du  8 novembre au 15 décembre 2017 à raison de deux séances de 2 heures de cours et 1 heure de travaux dirigés (exercices, exposés des étudiant-e-s) par semaine.


Basic homological algebra (chain complexes, homotopy, tensor product) and first notions of category theory (category, functor, monoidal category, monad).


       Bruno Vallette (Cours/TDs)

Back to main page

Last updates : July 18th, 2017