Throughout this talk I will use the following conventions and notations. I will primarily consider operads in the category of compactly generated Hausdorff topological spaces having the homotopy type of CW-complexes. When I refer to simplicial operads or operads in the category of posets, it will be understood that they can be converted to topological operads by taking geometric realization or ...

1 ∞-Operads 4 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Fibrations of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Cartesian Monoidal Structures . . . . . . . . . . ....

Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform operad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of view and the tools offered by the theory of opera...

We exhibit a parallel between Lawvere’s algebraic theories and operads; there is a common ancestor of both notions whose syntax is described by labeled trees, dummy variables corresponding to special labels. Operads, cyclic and braided operads alike arise as group objects in the category of compositional structures. Laplaza’s coherence theorem for “associativity not an isomorphism” is seen to c...

The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying algebraically the structures thus obtained (changes of bases, generating sets, presentations, morphisms, representations), to collect combinatorial informatio...

This open access book on higher category theory carefully develops the of infinity-operads in parallel to infinity-categories.

We describe combinatorial Hopf (co-)operadic models for the Swiss Cheese operads built from Feynman diagrams. This extends previous work of Kontsevich and Lambrechts-Voli\'c little disks operads.

1 ∞-Operads 4 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Fibrations of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Cartesian Monoidal Structures . . . . . . . . . . ....