An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them. The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields. The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

An operad is a mathematical object which allows us to encode the operations acting on categories of algebras. In this course, we will define the notion of operad together with many examples. We will then develop the homological algebra for operads leading to the Koszul duality theory. We will finish with the applications to the homotopy theory and open the doors to the deformation theory of algebraic structures.