Informally, an operad is a space of operations having one output and several inputs that can be composed. Each operad leads to the definition of category of algebras. This theory offers a tool to study situations wherein several operations interact with each others.
This lecture begins by presenting some elementary objects of algebraic combinatorics: combinatorial classes and combinatorial algebras. We introduce then (non-symmetric) operads and study some tools allowing to establish presentations by generators and relations of operads. Koszul duality in non-symmetric operads is an important part of this theory which shall be presented.
We end this lecture by reviewing some generalizations: colored operads, symmetric operads, and pros. We shall also explain how the theory of operads offers a tool to obtain enumerative results.
Here are the main presented notions: