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 collections and treelike structures. We introduce then rewrite systems on trees, (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 slides of the lecture (to be updated).
CM 1, January 20, 2021 : introduction about algebraic combinatorics : dendriform algebras, duplicial algebras, and pre-Lie algebras; First overview about operads.
CM 2, January 27, 2021 : collections; operations on collections; treelike structures.
CM 3, February 3, 2021 : syntax trees; rewrite systems on syntax trees.
CM 4, February 10, 2021 : proving termination and confluence of rewrite systems; prefix words encoding for syntax trees and termination invariants.
CM 5, February 17, 2021 : spaces of series and of polynomial on collections; operads; operads of paths, of permutations, and of rooted trees (pre-Lie operad); free operads; treelike expressions.
CM 6, February 24, 2021 : presentations of operads; proving presentations of operads through rewrite systems.
CM 7, March 3, 2021 : algebras over operads; Koszul dual of an operad; Koszulity.
CM 8, March 10, 2021 : two examples of constructions of combinatorial operads: operads from monoids (words) and operads from unitary magmas (decorated cliques).
At the end of each chapter, the above slides contain some exercises.