The Properadic Calculus

The operadic calculus developed over the past 20 years has now reached the status of a complete theory which is fruitfully used in many domains. It allows us to work out the homotopy properties of algebras, that is, algebraic structures with multiple inputs but one output. For instance, it encodes seminal notions like homotopy algebras and their infinity-morphisms, and it produces functorially such structures: homotopy transfer theorem, twisting procedure, and Koszul hierarchy. In this talk, I will survey the recent development of the properadic calculus which produces the similar tools and results but for algebraic structures made up of many inputs and many outputs, like the ones appearing in topological recursion, string topology, Poincaré duality and deformation theory.