#### Samuele Giraudo: Combinatorial realizations of algebraic structures (Slides, Enregistrement)

We focus on some interactions between combinatorics and various aspects of operad and clone theory. Given a variety of algebras (presented by generating operations and relations between some of their compositions), a natural question consists in building a realization of it. This consists in encoding the operations of the variety in such a way that two equivalent operations have the same representation, and providing an algorithm to compute the representation of the composition of several operations. In particular, we present a class of operads obtained from monoids and its counterpart in the context of clones. This gives rise among other to realizations of varieties of various classes of semigroups (commutative, left/right-regular bands, semilattices, etc.).

#### Guillaume Laplante-Anfossi: Operads and polytopes (Slides, Enregistrement)

We will start by introducing the notion of an operad, the connecting thread of the day, and we will give some examples of its appearance in various branches of mathematics. We will then study in more details the operad Ass, and we will show how it encodes (and generalizes!) the notion of an associative algebra. The latter behaving badly with respect to homotopy theory, we will try to replace it by another structure where the associativity relation holds only « up to homotopy ». Surprisingly enough, we will end up considering a family of simple geometric objects called « associahedra » or « Stasheff polytopes ». Finally, we will sketch the problem of the diagonal of the associahedra and we will present the interest of a particularly elegant family of realizations of these polytopes: the Loday realizations.

#### Maxime Lucas: Rewriting meets Homotopy (Slides, Enregistrement)

Weak algebraic structures are increasingly common, both in computer science (e.g. complete bicategories as models for the lambda calculus, oo-topoi as models for homotopy type theory) and in mathematical practice (e.g. Aoo or Eoo-algebras). The most well-known is probably the notion of monoidal category, which is a categorification of the notion of monoid. Nevertheless, finding a suitable weakening of an algebraic structure (encoded e.g. as an operad) is often complicated: while in good cases standard techniques exist, they often yield objects which are too large to be usable in practice.

In this talk, we start by motivating why it is natural to consider weakened versions of algebraic structures. We then explain the general setting in which categorification takes place, and what makes a weak structure the "right" categorification of a classical notion. Finally we show how to effectively compute small categorifications of structures given by generators and relations, using rewriting methods.

#### Damiano Mazza: Higher multicategories and programming languages (Slides, Enregistrement)

In this introductory talk we will see how the language of higher operads and, more generally, higher multicategories is useful to model the syntax and semantics of programming languages. In particular, we will see how, in this framework, the perspective introduced by Melliès and Zeilberger that "a type system is a functor" allows, in combination with suitable notions of fibration, the formulation of very general theorems about type systems and their properties.