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May 19-21, at Institut Henri Poincaré.

These three Federation Days will be devoted to the combinatorial, categorical and homotopic aspects of operad theory and their application to ongoing research at the interface of proof theory, programming language semantics, categorical algebra, mathematical physics and geometry.

Monday 19 May – All talks at **Amphitheater Hermite**

Tuesday 20 May – All talks at **Amphitheater Perrin** (at the Institut de Chimie Physique
just in front of the IHP and on the other side of the road)

Wednesday 21 May – All talks at **Amphitheater Darboux**

Monday 19 May | ||
---|---|---|

9h30-10h00 | Welcome | Tea, coffee and cakes at the ground floor lobby of the Institut Henri Poincaré |

10h00-11h00 | Martin Hyland | Operads and algebraic theories |

11h00-12h00 | Paul-André Melliès | Topological investigations on logical proofs |

12h00-14h00 | Lunch | |

14h00-15h00 | Ezra Getzler | Quasicategories and enrichment |

15h00-16h00 | Emily Riehl | The formal theory of adjunctions, monads, algebras, and descent |

16h00-16h30 | Coffee break | |

16h30-17h30 | Nicola Gambino | On operads, bimodules and analytic functors |

Tuesday 20 May | ||

9h30-10h30 | John Baez | Operads and the Tree of Life |

10h30-11h00 | Coffee break | |

11h00-12h00 | Benoit Fresse | Koszul duality, deformation complexes, and homotopy automorphisms of operads |

12h00-14h00 | Lunch | |

14h00-15h00 | Matthieu Anel | Enriching algebras over coalgebras and operads over cooperads |

15h00-16h00 | Muriel Livernet | Non-formality of the Swiss-cheese operad |

16h00-16h30 | Coffee break | |

16h30-17h30 | Joachim Kock | Combinatorial Dyson-Schwinger equations, polynomial functors,
and operads of Feynman graphs |

Wednesday 21 May | ||

9h30-10h30 | Gabriele Vezzosi | Recent directions in derived geometry |

10h30-11h00 | Coffee break | |

11h00-12h00 | Eric Finster | Opetopic Diagrams as a Language for Higher Categorical Proofs |

12h00-14h00 | Lunch | |

14h00-15h00 | Philippe Malbos | Higher-dimensional algebroids and Koszulity |

15h00-16h00 | Frédéric Chapoton | A combinatorial source of many operads |

16h00-16h30 | Coffee break |

**Matthieu Anel** (ETH Zurich, Switzerland)

*Enriching algebras over coalgebras and operads over cooperads*

I will first explain how to enriched associative algebras over coassociative coalgebras and show how this enrichment provides a new presentation of the bar-cobar constructions. Then I will sketch how to extend these results for (co)algebras over a Hopf operad. We will deduce in particular an enrichment of operad over cooperads and a new approach to the operadic bar-cobar constructions. This is a work in progress with A. Joyal.

**John Baez** (U. C. Riverside)

*Operads and the Tree of Life*

Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists seek to reconstruct the most likely phylogenetic tree describing how these species evolved from earlier ones. In fact, phylogenetic trees are operations in an operad, the “phylogenetic operad”, which plays a universal role in the study of branching Markov processes. To understand this operad, and more generally the relation between operads and trees, we use the fact that operads are themselves the algebras of a (typed) operad. This is joint work with Nina Otter and Todd Trimble.

**Frédéric Chapoton** (Institut Camille Jordan, Université Claude Bernard Lyon 1)

*A combinatorial source of many operads*

Using a simple combinatorial recipe, one can build many different examples of non-obvious non-symmetric operads. For each one of them, there are then two associated challenges : find the dimensions and obtain a presentation. Already in the simplest case of two colors, there are interesting combinatorial objects involved. Joint work with Samuele Giraudo.

**Eric Finster** (Université Paris Diderot)

*Opetopic Diagrams as a Language for Higher Categorical Proofs*

Recent advances in type theory have demonstrated interesting new links between formal languages, formal proof assistants and ideas from higher category theory. The identity types of Martin-Lof type theory give us a systematic logical formalism for reasoning synthetically about weak infinity groupoids, or equivalently, homotopy types.

On the other hand, the problem of applying these ideas to higher categories, in place of higher groupoids, remains open. Here I will demonstrate a prototype proof assistant for higher category theory based on the opetopic model of higher categories. The proof assistant employs a diagrammatic syntax based on the opetopic diagrams of Joyal, Kock, Batanin and Mascari and implements a definition of opetopic higher category due to Thorsten Palm.

**Benoit Fresse** (Laboratoire Painlevé, Université Lille 1)

*Koszul duality, deformation complexes, and homotopy automorphisms of operads*

Homotopy automorphism spaces of operads encode the internal symmetries of homotopy categories of algebras governed by operads. I will explain the definition of these objects in a first introductory part of my talk. In a second step, I will explain the general definition of spectral sequences to compute these homotopy automorphism spaces, and show how the method applies in the E_n-operad case.

**Nicola Gambino** (University of Leeds)

*On operads, bimodules and analytic functors*

In spite of their very different origins, the theory of operads and the theory of analytic functors are deeply connected. The aim of this talk is to give an overview of these connections and present some recent results, obtained in collaboration with André Joyal. In particular, we have shown that operads, operad bimodules and bimodule maps form a cartesian closed bicategory. The proof involves some results on bicategories of bimodules which are of independent interest.

**Ezra Getzler** (Northwestern University)

*Quasicategories and enrichment*

One of the key theorems in the theory of quasicategories is the construction of the infinity-groupoid G(C) of invertible arrows of a quasicategory C. In this talk, we develop an enriched version of this construction, with applications to higher analytic stacks (and derived stacks). The main observation is that in the absence of colimits, it is better to work with a certain equivalent infinity-groupoid, which was introduced in the topological setting at around the same time (late 90s) by Rezk in his work on complete Segal spaces.

This work is part of our collaboration with Kai Behrend on the construction of analytic higher stacks of twisted complexes of vector bundles over a compact complex manifold.

**Martin Hyland** (University of Cambridge)

*Operads and algebraic theories*

A natural setting for the notion of coloured operad is provided by a bicategory determined by the notion of symmetric monoidal category. Joyal's theory of species lives in the same bicategory. Other bicategories support the notion of non-symmetric operads on the one hand and the standard notion of algebraic theory on the other. I shall explain the abstract mathematics behind such bicategories and their use in moving between different flavours of theory.

The general theory is not at all restricted to the standard examples. I shall illustrate the possibilities by constructing a notion of theory containing operadic and purely algebraic elements. This new kind of theory is the basis for ongoing foundations investigations concerning the differential lambda calculus.

**Joachim Kock** (Universitat Autònoma de Barcelona)

*Combinatorial Dyson-Schwinger equations, polymomial functors, and operads of Feynman graphs*

After briefly recalling the Connes-Kreimer Hopf algebra of Feynman graphs and the Butcher-Connes-Kreimer Hopf algebra of trees, as they appear in BPHZ renormalisation in Quantum Field Theory, a main point of this talk is to explain the relationship between the two via an intermediate bialgebra of P-trees, for P a certain polynomial functor of primitive graphs, and to outline some categorical interpretations resulting from this viewpoint. The combinatorial Dyson-Schwinger equations of Bergbauer-Kreimer take the form of polynomial fixpoint equations in groupoids, X = 1 + P(X); the solution X (which is accordingly a W-type in the sense of type theory) consists of certain nested graphs, which are the P-trees, appearing as the operations of $\bar P$, the free monad on P. The so-called overlapping divergences, a main subtlety of BPHZ renormalisation, are interpreted as the relations defining the operad of graphs as a quotient of $\bar P$. In Quantum Chromodynamics one is interested in certain truncations of the DSEs, required to generate sub-Hopf algebras. In the abstract setting, truncation is interpreted as monomorphic natural transformations between polynomial endofunctors, and the sub-hopf condition is satisfied by the cartesian natural transformations.

**Muriel Livernet** (Institut Galilée, Université Paris Nord)

*Non-formality of the Swiss-cheese operad*

I will review in this talk the notion of formality for operads and some classical results. I will then introduce the Swiss-cheese operad and explain the differences between the Swiss-cheese operad and the little disk operad. Finally I will prove the non-formality of the swiss-cheese operas.

**Philippe Malbos** (Institut Camille Jordan, Université Claude Bernard Lyon 1)

*Higher-dimensional algebroids and Koszulity*

We define linear polygraphs as higher dimensional linear rewriting systems for presentations of algebras, generalizing the notion of noncommutative Gröbner bases. They are constructed on the notion of category enriched in higher-dimensional vector spaces. We will show how to construct polygraphic resolutions for algebras, starting from a convergent presentation, and how to relate these resolutions with the Koszul property. This is a joint work with Eric Hoffbeck and Yves Guiraud.

**Paul-André Melliès** (CNRS, Université Paris Diderot)

*Topological investigations on logical proofs*

The purpose of this introductory talk will be to illustrate how operad theory helps us to understand the mathematical structure of proofs and programs. More than 40 years ago, Lambek observed that the free cartesian closed category is a category of types and terms of the simply typed lambda-calculus. From this follows that every cartesian closed category defines a proof-invariant modulo execution. Somewhat surprisingly, the very same functorial principle reappeared in knot theory in the early 1990s with the notion of ribbon (or tortile) category. Indeed, just as in the case of cartesian closed categories, every ribbon category defines a knot-invariant modulo topological deformation. This similarity between proof theory and knot theory leads to the notion of dialogue category which lies at the frontier of the two fields. I will explain how to understand the notion of dialogue category from a purely 2-categorical point of view inspired by operad theory, and how to establish in this way that the free dialogue category is a category of dialogue games and innocent strategies. I will conclude with a series of somewhat mysterious connections to cubical sets and to the little disk operad.

**Emily Riehl** (Harvard University)

*The formal theory of adjunctions, monads, algebras, and descent*

We develop a theory of “algebraic structures” - more precisely, a theory of adjunctions, monads, their algebras, and monadic descent - in which both the definitions and the proofs are simultaneously interpretable in any (strict) 2-category or (∞,2)-category admitting a certain class of weighted limits. Here (∞,2)-category means a simplicially enriched category whose hom-spaces are quasi-categories. These contexts are related: 2-category theory is a special case of (∞,2)-category theory, and conversely, any (∞,2)-category has a “homotopy 2-category.” In the (∞,2) context, the adjunctions are “homotopy coherent”; moreover, we show that any adjunction in the homotopy 2-category of any (∞,2)-category lifts to a homotopy coherent adjunction. We prove the monadicity theorem, again simultaneously in both contexts, and apply it to develop the theory of monadic descent. This is joint work with Dominic Verity.

**Gabriele Vezzosi** (Université Paris Diderot)

*Recent directions in derived geometry*

We will give an idea of derived algebraic geometry and sketch a number of more or less recent directions, including derived symplectic geometry, derived Poisson structures, quantizations of moduli spaces, and derived logarithmic geometry.

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