**Mini-cours:** Greg Arone (Stockholm University)

**Title:**Goodwillie calculus.

**Abstract:**Calculus of functors is a categorification of differential calculus of Newton and Leibnitz. In this calculus, functors between infinity categories play the role of differentiable functions. It turns out that many concepts from differential calculus have categorical analogues. Thus, one can define the derivatives of a functor, Taylor approximations of a functor, and so forth. This philosophy, which was first conceived by Goodwillie in the 1980-ies, and then developed further by Michael Weiss, has had numerous applications in algebraic and geometric topology. In these lectures we will outline the main results of the theory and survey some applications. If time permits, we will also survey some recent developments.

### Conférenciers invités

#### Sylvain Douteau (Université Paris Cité)

**Title :**Generalized intersection cohomology and homotopy Deligne decomposition

**Abstract :**Understanding intersection cohomology from a homotopy theoretic viewpoint is an old problem, dating back to its introduction by Goresky and MacPherson. There are at least two distinct phenomena that one would want to study from this perspective. Whether intersection cohomology is representable in some sense, and whether it can be axiomatically described, à la Eilenberg-Steenrod, leading to generalized intersection cohomology theories. While both questions have received partial answers, a unifying framework is still lacking.

#### Benoit Fresse (UNiversité de Lille)

**Title: **The computation of mapping spaces of rational E_n-operads through graph complexes, with a view towards applications in the embedding calculus

#### Goeffroy Horel (Université Sorbonne Paris Nord)

**Title:** Hypercommutative algebra structures on Kähler and Calabi-Yau manifolds**Abstract:** Any Batalin–Vilkovisky algebra with a homotopy trivialization of the BV-operator gives rise to a hypercommutative algebra structure at the cochain level which, in general, contains more homotopical information than the hypercommutative algebra introduced by Barannikov and Kontsevich on cohomology. In this talk, I will explain how to use the purity of mixed Hodge structures to prove formality of certain

hypercommutative algebras associated to Kähler and Calabi-Yau manifolds.

This can be viewed as a generalization of the famous result of Deligne-Griffiths-Morgan-Sullivan proving formality of the commutative algebra of differential forms on Kähler manifolds. This is joint work with Joana Cirici.

#### Muriel Livernet (Université Paris Cité)

**Title:** A walk on cubes and simplices.

**Abstract:** The aim of the talk is to compare different models for infinity categories. The talk is presented in the form of a stroll, leading us to homotopy 3-types.

#### Delphine Moussard (Aix-Marseille Université)

**Title:** Multisections of smooth and PL closed manifolds

**Abstract:** We consider a notion of multisection for closed manifolds which generalizes Heegaard splittings of 3-manifolds and Gay-Kirby trisections of smooth 4-manifolds: a multisection of a closed manifold is a decomposition into 1-handlebodies, where any subcollection meets along a 1-handlebody, except the global intersection which is a closed surface. We will discuss existence and uniqueness of such decompositions and see how they can be represented by diagrams on closed surfaces. Joint work with Fathi Ben Aribi, Sylvain Courte and Marco Golla.

#### Julia Semikina (Université de Lille)

**Title:**Cut-and-paste K-theory of manifolds and cobordisms

**Abstract**

**:**The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope P in R^n, one can cut P into a finite number of smaller polytopes and reassemble these to form Q. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut and paste relation for manifolds called the SK-equivalence ("schneiden und kleben" is German for "cut and paste").

#### Niall Taggart (Université d'Utrecht)

**Title:** Power operations and free spectral Lie algebras**Abstract:** Power operations are essential in making cohomology an effective tool for studying spaces, with examples including Steenrod operations on ordinary cohomology, Dyer-Lashof operations on iterated loop spaces, and Adam's operations in complex K-theory. In this talk I will describe an approach to study certain power operations on free spectra Lie algebras at chromatic height one. This approach hinges on the relationship between calculus of functors and splitting results of Cohen, Moore and Neisendorfer. This is joint work (in progress) with Blans, Boyde and Heuts.

#### Christine Vespa (Aix-Marseille Université)

**Title:**Polynomial functors associated with beaded open Jacobi diagrams

**Abstract:**The Kontsevich integral is a very powerful invariant of knots, taking values is the space of Jacobi diagrams. Using an extension of the Kontsevich integral to tangles in handlebodies, Habiro and Massuyeau construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. The category A has a subcategory equivalent to the linearization of the opposite of the category of finitely generated free groups, denoted by

**gr**^{op}. By restriction to this subcategory, morphisms in the linear category

**A**give rise to interesting contravariant functors on the category

**gr**, encoding part of the composition structure of the category

**A**.

**A**from 0. In particular, she obtains a family of polynomial functors on

**gr**^{op} which are

*outer functors*, in the sense that inner automorphisms act trivially.

**gr**^{op} which are no more outer functors. Our approach is based on an equivalence of categories given by Powell. Through this equivalence the previous polynomial functors correspond to functors given by beaded open Jacobi diagrams.

### Exposés sur proposition:

#### Alexis Aumonier (Cambridge)

**Title: **Scanning the moduli of embedded smooth hypersurfaces**Abstract: **The Hilbert scheme of a variety X is an important moduli space in algebraic geometry: it parameterises closed subschemes of X. It is usually too big and badly behaved to be considered as a whole. Instead, I will focus in this talk on the locus parameterising smooth hypersurfaces. I will explain how one can compute a bit of its cohomology using tools from algebraic topology akin to scanning methods, and reveal a connection to moduli spaces of high dimensional manifolds as studied by Galatius and Randal-Williams.

Although algebro-geometrically sounding, this talk will be given for algebraic topologists by an algebraic topologist.

#### Merlin Christ (Paris)

**Title: **Complexes of stable infinity-categories.**Abstract:** We introduce a notion of lax additive (infinity,2)-category, categorifying the notion of additive 1-category. One can define and study complexes valued in any lax additive (infinity,2)-category, and consider these as categorified chain complexes. We refer to complexes valued in the lax additive (infinity,2)-category of (presentable) stable infinity categories as categorical complexes. We explicitly describe a categorified totalization construction, which produces a categorical complex from a categorical multicomplex. Special cases include the totalizations of commutative squares or higher cubes of stable infinity categories, which give rise to many examples of such categorical complexes. These constructions describe additive aspects of the conjectural/emerging subject of categorified homological algebra. This talk is based on joint work with T. Dyckerhoff and T. Walde.

#### Marie-Camille Delarue (Paris)

**Titre:** Using scanning methods to prove the Barratt-Priddy-Quillen theorem**Abstract:** The Barratt-Priddy-Quillen theorem establishes a homology equivalence between the group completion of the union of the symmetric groups and the infinite loop space of the sphere spectrum. We study the symmetric groups using paths between configurations embedded in R^\infty and use scanning methods to provide another proof of this result. This method may be extended to the computation of the stable homology of other families of groups in future works.

#### Coline Emprin (Paris)

**Title:** Kaledin classes and formality criteria**Abstract:** A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, an operad, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. Kaledin classes were introduced as an obstruction theory characterizing the formality of associative algebras over a characteristic zero field. In this talk, I will present a generalization of Kaledin classes to any coefficients ring and to other algebraic structures (encoded by operads, possibly colored, or by properads). I will prove new formality criteria based on these classes and give applications.

#### Antoine Feltz (Strasbourg)

**Title:** Polynomial functors over the categories FId of injections with colours**Abstract: **The functors over the category FI of finite sets and injections appear naturally in different contexts. They intervene especially in the theory of twisted commutative algebras (TCA), or in the study of representation stability initiated by Church, Ellenberg and Farb which applies, for example, to the cohomology of configuration spaces. Djament and Vespa showed that the representation stability can be expressed in terms of polynomial functors over FI (called strong polynomiality). They introduced also another notion of polynomiality better for stable phenomenon (called weak polynomiality).

There exist generalizations of the category FI, denoted FId, where we add a choice of colours among d possible on the complement of the image of each injection. The functors over these categories intervene notably in the work of Sam and Snowden on modules over the free TCA, and in the work of Ramos on cohomology of graph configuration spaces. As the categories FId do not have initial object anymore for d > 1, they come out of the framework of Djament and Vespa.

In this talk we will introduce different notions of polynomial functors over the categories FId, and we will illustrate how they turn out to be harder to study than on FI. For example, the standard projective functors on FI are strongly polynomial. This is not the case anymore on FId for d > 1, which prompts us to study their polynomial quotients. Moreover, while the weak polynomial functors of degree 0 over FI are the constant functors, we give a description of those on FId which form a more complex category.

#### Owen Garnier (Amiens)

**Title :** Homology Computations for complex braid groups with the Dehornoy-Lafont order complex

**Abstract.** In addition of its topological origin as a fundamental group, the braid group on n strands also follows a rigid combinatorial behavior, which was first noticed by F.A.Garside

in 1969. This behavior was later axiomatized in the notion of Garside monoid by Dehornoy and Paris, and generalized to all Artin groups of spherical type. Among other results, a Garside monoid for a group B gives rise to several free resolutions of the trivial module over B. One of these resolutions is the so-called Dehornoy-Lafont order complex. This complex has a relatively small number of cells, which makes it rather efficient in practical computations.

More recently, from the study of complex braid groups arose the need to consider not only Garside monoids, but also Garside categories. This raises the question of adapting the Dehornoy-Lafont complex to the categorical context. In this talk we will first explain the relations between the homology of a group, that of a groupoid to which it is equivalent, and that of a generating subcategory of such a groupoid. We will then detail the construction of the Dehornoy-Lafont complex for a Garside category, before discussing a good heuristic method for minimizing the number of cells.

If time permits, we will give an application to the case of the complex braid group B(G31), which is studied through its associated Garside category, and which was not directly covered by previous approaches.

#### Samuel Lavenir (Lausanne)

**Title:** Persistent rational homotopy theory**Abstract:** Rational homotopy theory aims at studying homotopy types by disregarding the torsion phenomena that inevitably occur in homotopy theory. Its power comes from the ability to model homotopy types using algebraic invariants such as (differential graded) commutative algebras. The use of minimal models - as developed by Sullivan - benefits the theory by allowing explicit computations to be performed. In this talk, we outline how the ideas of persistence can be applied to the context of rational homotopy theory. After describing how obstruction theory can be used to compute explicit minimal models for persistent (differential graded) commutative algebras, we will describe how this construction hinted the idea of a novel model structure on the category of persistent cochain complexes. This structure is described by explicit sets of generating cofibrations with respect to which persistent minimal models become cell complexes. By transfer along free-forgetful adjunctions, we obtain cofibrantly generated model structures on various categories of persistent differential graded algebras. This is joint work with Kelly Maggs and Kathryn Hess.

#### Silvère Nédélec (Nantes)

**Title : **A special case of rewritting theory for properads.**Abstract :** In the context of operads, rewriting theories allow us to study operads and their Koszulness, but in the context of properads, no such theory exists yet. The goal of this talk is to study the special case of properads where the operations are generated by an associative product and a coassociative coproduct, and see if in this special case, we can conclude on an equivalence between confluence and Koszulness. The case of confluent properads of that type is quite simple since most of them are already studied in the literature, but for the non-confluent case, we need computer assistance by Sagemath to compute the dimensions of free properads and certain ideals in properads.

#### Emile Oléon (Paris)

**Title:** Hypercubical manifolds and quaternion group in Cubical Type Theory**Abstract:** Cubical Type Theory is a slight variant of Homotopy Type Theory in which types can depend on ”directions” and hence be cubical in a geometrical sense. This setting will allow us to define the hypercubical manifold, a space first studied by Poincar´e during his quest to define homology spheres, in a concise (and usable) way. We will also show how we can use the join in homotopy type theory to construct higher dimensional version of this space, leading to better and better approximations of the homotopy type of a K(Q, 1) as the dimension increase, in the sense that the inductive limit will be a delooping of Q. This will lead to a definition in homotopy type theory of a delooping of Q which have a cellular structure (as opposed to classical models of K(G, 1) in HoTT), hence opening the way to a computation of the cohomology of Q in a synthetic setting (and formalised by proof assistants).

### Groupe de travail doctorants

La journée du lundi 23 octobre est consacrée à un groupe de travail des doctorants sur le thème des limites homotopiques, organisé par Nicola Carissimi et Marvin Verstraete.