Rencontre 2024 de Topologie algébrique (Réseau Thématique)

21 oct. 2024, 09:00 → 25 oct. 2024, 17:00 Europe/Paris

Amphi Schwartz (IMT)

(crédit : DCCE Université Toulouse III - Paul Sabatier)

Inscription avant le 30 septembre !!!

^{(avant le 10 septembre si demande de financement)}

La réunion annuelle 2024 a lieu à Toulouse, à l'institut de mathématiques, du 21 au 25 octobre 2024. La journée du 21 octobre est consacrée à un groupe de travail de doctorants du RT pour préparer le mini-cours (programme).

Mini-cours intitulé "Free and equal higher categories" :

Viktoriya Ozornova (institut Max Planck à Bonn, Allemagne)

 

Oratrices et orateurs :
Clément Dupont (université de Montpellier)

Niels Feld (université de Rennes)

Mario Fuentes (université de Toulouse)

Eric Hoffbeck (université Sorbonne Paris Nord)

Inbar Klang (université d'Amsterdam, Pays-Bas)

Clover May (NTNU, Trondheim, Norvège)

Morgan Rogers (université Sorbonne Paris Nord)

Mohamed Ayadi (université d'Angers)

Oisin Flynn Connolly (université Sorbonne Paris Nord)

Jean Douçot (université de Lisbonne)

Léonard Guetta (université d'Utrecht)

Léo Hubert (université de Marseille)

Samuel Lerbet (université de Grenoble)

Hugo Pourcelot (université de Florence)

Francesca Pratali (université Sorbonne Paris Nord)

Paula Verdugo (institut Max-Planck à Bonn)

Marvin Verstraete (université de Lille)

 

 

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Organisation et comité scientifique

Comité scientifique : Christian Ausoni (université Sorbonne Paris Nord), Julie Bergner (university of Virginia, Etats-Unis), Frédéric Chapoton (université de Strasbourg), Joana Cirici (universitat de Barcelona, Espagne), Frédéric Déglise (ENS de Lyon), Najib Idrissi-Kaïtouni (université Paris Cité), Muriel Livernet (université Paris Cité), Jérôme Scherer (Ecole Polytechnique Fédérale Lausanne, Suisse), Antoine Touzé (université de Lille).

Organisateurs : Ricardo Campos, Joan Millès et Joost Nuiten (organisateurs locaux),  Antoine Touzé (université de Lille, responsable du réseau thématique).

Organisateurs du groupe de travail des doctorants : Jiaqi Fu et Francesca Pratali.

 

 

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Sponsors

L'organisation de la conférence annuelle du RT est soutenue par :

• le RT 2171 "Topologie algébrique et géométrique" du CNRS
• l'institut de mathématiques de Toulouse et
• le labex CIMI
• la fédération occitane de mathématiques OcciMath.

lundi 21 octobre

09:30 Accueil pour le groupe de travail préparatoire au mini-cours (Higher categories in an éclair)

1 Model categories

Orateur: José São João

2 2-categories I

Orateur: Benachir El Allaoui

12:00 Repas à l'upsidum

3 Simplicial sets and quasi-categories

Orateur: Tommy-Lee Klein

4 2-categories II

Orateur: Nicolas Guès

16:00 Pause café

5 Models of (oo, 1)-categories

Orateur: Paul Laubie

6 Bonus: question session

mardi 22 octobre

09:40 Accueil et pause café

7 Little disks and tangential basepoints: an invitation to logarithmic geometry

Various maps arising in topology look like they should come from algebraic geometry, if only one could allow them to take values « at infinity ». This is the case for the little disks operad, whose underlying spaces have the homotopy type of the configuration spaces of points on the affine line, but whose operadic maps are not algebraic in any obvious way. I will explain how to solve this problem, thereby realizing Beilinson’s strategy of a « purely algebraic » proof of the formality of little disks, using logarithmic geometry. Perhaps surprisingly, one needs a non-standard notion of morphism in logarithmic geometry, dubbed « virtual morphism » by Howell. That notion also allows to realize Deligne’s tangential basepoints, which are crucial in Grothendieck—Teichmüller theory, as points in the categorical sense. This is joint work with Erik Panzer and Brent Pym.

Orateur: Clément Dupont (Université de Montpellier)

8 Mini-cours - Free and equal higher categories

The main objective of this series is to discuss the relationship of strict and (various variants of) weak higher categories. The beginning is the case of usual categories being embedded into $(\infty,1)$-categories. We will discuss this embedding in different models for $(\infty,1)$-categories, alongside with a reminder on these models. Once the case $n=1$ is somewhat understood, we will move on to a similar analysis for $n=2$. For $n>2$, the situation is much more open, and I will describe the state of art there.

Orateur: Viktoriya Ozornova

12:55 Repas à l'upsidum

9 Isomonodromic deformations of irregular connections and wild mapping class groups

Via the Riemann-Hilbert correspondence, character varieties
can be seen as moduli spaces of monodromy data of meromorphic
connections with regular singularities on a Riemann surface. Varying the
curve with marked points, this leads to isomonodromic deformations and
to the well-known mapping class group actions on character varieties.

This story admits a far-reaching generalization if we do not restrict to
regular singularities: we now have to consider moduli spaces of
generalized monodromy data, giving rise to wild character varieties.
There are then new deformation parameters: the irregular types of the
connections at their singularities.

We will discuss some recent work, in collaboration with P. Boalch, G.
Rembado, and M. Tamiozzo, where we define and study the spaces of these
deformation parameters and their fundamental groups, yielding wild
mapping class groups. They act on the wild character varieties by
Poisson automorphisms, opening the door to many questions about these
wild dynamics.

Orateur: Jean Douçot

10 Equivariant free loop spaces and equivariant algebraic K-theory

I will give an introduction to topological Hochschild homology via free loop spaces and configuration spaces, and talk about its relationship with algebraic K-theory. I will then discuss what happens in the presence of a finite group action. This will touch upon joint work with Adamyk, Gerhardt, Hess, and Kong, and joint work in progress with Chan and Gerhardt.

Orateur: Inbar Klang

16:25 Pause café

11 Corecognition for iterated suspensions

In this talk, I will present joint work with José Moreno-Fernández and Felix Wierstra on coalgebras in topological spaces. We will construct the comonad associated to a topological operad and we will sketch the proof of a recognition principle for iterated suspensions as coalgebras over the little cubes operad. These statements are Eckmann-Hilton dual to May's foundational results on iterated loop spaces. Afterwards we shall discuss some applications to homotopy operations.

Orateur: Oisin Flynn Connolly

mercredi 23 octobre

12 Complete (curved) Lie algebras as models of spaces

The rational homotopy type of simply connected spaces is fully captured by its Quillen model, a differential graded Lie algebra constructed from the space. Conversely, any positively graded differential Lie algebra can be "realized" as a topological space, with rational homotopical and homological invariants preserved by these two functors.

However, these constructions are inherently limited to connected and simply connected spaces. To remove these constraints, we must move to the category of complete Lie algebras. Within this category, there exists a cosimplicial object that gives rise to a pair of adjoint functors between the categories of complete Lie algebras and topological spaces.

In this talk, we will explore the
construction of this pair of functors and some important properties. Concretely, we will show that composing both of them results in the Bousfield-Kan $\mathbb{Q}$-completion. Additionally, we will discuss how this framework can be extended to curved Lie algebras, leading to a "base pointless" theory.

Orateur: Mario Fuentes

13 An example of twisted bialgebras of finite topological quandles

In this presentation, I will discuss the classification method for finite topological quandles for a given cardinality n. As an application, we classify finite topological quandles with up to 4 elements. Then in a second step, I will talk about twisted bialgebra structures within the realm of finite topological quandle species, one of the first kind and one of the second kind. The obstruction for the structure to match the double twisted bialgebra axioms is explicitly described.

These findings are the result of collaboration with Dominique Manchon.

Orateur: Mohamed Ayadi

11:00 Pause café

14 Mini-cours - Free and equal higher categories

The main objective of this series is to discuss the relationship of strict and (various variants of) weak higher categories. The beginning is the case of usual categories being embedded into $(\infty,1)$-categories. We will discuss this embedding in different models for $(\infty,1)$-categories, alongside with a reminder on these models. Once the case $n=1$ is somewhat understood, we will move on to a similar analysis for $n=2$. For $n>2$, the situation is much more open, and I will describe the state of art there.

Orateur: Viktoriya Ozornova

12:55 Repas à l'upsidum

15 Motivic stable cohomotopy and unimodular rows

Unimodular rows are fundamental objects in algebraic K-theory as they classify stably free projective modules. Geometrically, they correspond to morphisms to punctured affine space and may thus be thought of as an algebro-geometric analogue of maps to spheres in topology. The latter give rise to Borsuk's cohomotopy groups, a counterpart to which was constructed purely algebraically by van der Kallen in the 1980s using certain orbit sets of unimodular rows. Nowadays, using the homotopy theory of schemes, Borsuk's construction may be followed more directly to produce the motivic cohomotopy groups of Asok and Fasel. We prove that these two approaches of “algebraic” cohomotopy, namely van der Kallen's and Asok–Fasel's, coincide in that there is a natural map from orbit sets of unimodular rows to motivic cohomotopy groups which is a group isomorphism in the dimension range where both structures exist.

Orateur: Samuel Lerbet (Institut Fourier, Université Grenoble Alpes)

16 Modules of equivariant Eilenberg--MacLane spectra

Cohomology with $\mathbb{Z}/p$-coefficients is represented by a stable object, an Eilenberg--MacLane spectrum $H\mathbb{Z}/p$. Classically, since $\mathbb{Z}/p$ is a field, any module over $H\mathbb{Z}/p$ splits as a wedge of suspensions of $H\mathbb{Z}/p$ itself. Equivariantly, cohomology and the module theory of $G$-equivariant Eilenberg--MacLane spectra are much more complicated.
For the cyclic group $G=C_p$ and the constant Mackey functor $\underline{\mathbb{Z}}/p$, there are infinitely many indecomposable $H\underline{\mathbb{Z}}/p$-modules. Previous work together with Dugger and Hazel classified all indecomposable $H\underline{\mathbb{Z}}/2$-modules for the group $G=C_2$. The isomorphism classes of indecomposables fit into just three families. By contrast, we show for $G=C_p$ with $p$ an odd prime, the classification of indecomposable $H\underline{\mathbb{Z}}/p$-modules is wild. This is joint work in progress with Grevstad.

Orateur: Clover May

16:25 Pause café

17 Perverse homotopy heart of stable motivic homotopy and Milnor-Witt cycle modules

In the nineties, Voevodsky proposed a radical unification of algebraic and topological methods. The amalgam of algebraic geometry and homotopy theory that he and Fabien Morel developed is known as motivic homotopy theory. Roughly speaking, motivic homotopy theory imports methods from simplicial homotopy theory and stable homotopy theory into algebraic geometry and uses the affine line to parameterize homotopies. Voevodsky developed this theory with a specific objective in mind: prove the Milnor conjecture. He succeeded in this goal and won the Fields Medal for his efforts in 2002.
In this talk, I will present an ongoing project in collaboration with Frédéric Déglise and Fangzhou Jin where we realize Ayoub's conjectural program showing that the heart of the motivic stable homotopy category over appropriate base schemes can be related to a suitable version of relative Milnor-Witt modules.

Orateur: Niels Feld

jeudi 24 octobre

18 Integration along the fiber for gebras over dioperads

Given a monoidal adjunction and a certain orientation datum on the right adjoint F, I will explain how to transport gebras over dioperads along F, via endowing this functor with a shifted Frobenius monoidal structure. This procedure generalizes so-called integration along the fiber, which is for instance the case when F is the pushforward of a projection X x M --> X, with M a closed oriented manifold.

Applied to the dioperad of shifted Lie bialgebras, the above construction can be regarded as a toy model of a conjectural AKSZ construction for shifted Poisson target (in the spirit of Pantev-Toën-Vaquié-Vezzosi's theorem on shifted symplectic structures). Time permitting, I shall discuss this application and justify the previous claim.

Orateur: Hugo Pourcelot (LAGA)

19 Deformation theories and pre-Lie algebras with divided powers in positive characteristic

An important result in deformation theory over a field of characteristic 0 asserts that every deformation problem can be controlled by a differential graded Lie algebra. More precisely, every solution of a deformation problem over a field of characteristic 0 can be seen as a Maurer-Cartan element in some dg lie algebra $L$. The isomorphisms classes of deformation problems are also in correspondence with the classes of Maurer-Cartan elements under the action of the integration of $L$ into a Lie group called the gauge group. In 2015, Dotsenko-Shadrin-Vallette developed the deformation theory controlled by pre-Lie algebras, which are examples of Lie algebras, over a field of characteristic 0. They showed that the gauge group and its action on the Maurer-Cartan elements can be expressed in terms of pre-Lie operations. In this talk, we generalize such results in the positive characteristic context by developing a deformation theory controlled by pre-Lie algebras with divided powers. We also prove that this deformation theory satisfies a Goldman-Millson theorem which is valid over the integers. We apply this deformation theory to the computation of operad morphisms up to homotopy from the cobar construction of an augmented cooperad to an augmented operad.

Orateur: Marvin Verstraete

11:00 Pause café

20 Mini-cours - Free and equal higher categories

The main objective of this series is to discuss the relationship of strict and (various variants of) weak higher categories. The beginning is the case of usual categories being embedded into $(\infty,1)$-categories. We will discuss this embedding in different models for $(\infty,1)$-categories, alongside with a reminder on these models. Once the case $n=1$ is somewhat understood, we will move on to a similar analysis for $n=2$. For $n>2$, the situation is much more open, and I will describe the state of art there.

Orateur: Viktoriya Ozornova

12:55 Repas à l'upsidum

21 Homotopical Dold-Kan Correspondences

Grothendieck's theory of test categories allows to characterize small categories with the property that an appropriate localization of their categories of presheaves modelize the homotopy category of spaces. Any test category then allows to do homotopy theory just as well as traditional simplicial sets can.
However, simplicial sets exhibit other niceties. Among them is the Dold-Kan correspondence: simplicial abelian groups also form a model for homology types, after an appropriate localization.
In this talk, we will see how we can hunt for homotopical Dold-Kan correspondences for presheaves in abelian groups over test categories and we will give some examples, including Joyal's category Theta.

Orateur: Léo Hubert

22 Rectification of operadic left fibrations

By a result of Heuts-Moerdijk, the oo-category of simplicial diagrams on the nerve of a discrete category A is equivalent to that of left fibrations over the nerve of A. This is an instance of the well known Grothendieck-Lurie straightening-unstraightening theorem.
In this talk, we will explain how one can generalize this result to the operadic case. More specifically, by working with the dendroidal formalism we show how, given any discrete operad P, one can functorially rectify an operadic left fibration over the dendroidal nerve of P and obtain a simplicial algebra on P. After explaining how this extends an analogous functor for categories, we prove that it establishes an adjunction of oo-categories between operadic left fibrations over the nerve of P and simplicial P-algebras. It is a work in progress that it establishes an equivalence of oo-categories, which would be a first step towards operadic straightening-unstraightening!
If time permits, we will conclude the exposition by presenting possible future applications.

Orateur: Francesca Pratali (Université Sorbonne Paris Nord)

16:25 Pause café

23 Monoids and (1-)toposes

My work revolves around monoids acting on things, especially in the context of (1-)topos theory. As such, it is through this lens that I view the ingedients of algebraic geometry, and I will tell you about this perspective. From a research perspective, this work hasn't gone very far yet; I merely intend to sketch out for you a different way of setting up the big picture.
Since (Grothendieck) 1-toposes probably won't otherwise have been discussed much this week, I'll take some time to reflect on the basic properties and constructions of these objects.

Orateur: Morgan Rogers

19:30 Repas de conférence

Le site du resto

vendredi 25 octobre

24 Double categories as a model for (oo,1)-categories

In this talk, I will present a joint work with Lyne Moser where we prove that double categories model (oo,1)-categories. More precisely, we equip the category of double categories with a model category structure and show that it is Quillen equivalent to the Rezk model category structure on bisimplicial sets.
I will also explain how this model of (oo,1)-categories is related to other "categorical" models, such as categories internal to simplicial sets due to Horel, relative categories due to Barwick and Kan and 2-categories due to Pavlov.
Finally, I will present some conjectural nice features of this model of (oo,1)-categories, which are motivating on their own.

Orateur: Léonard Guetta

25 On the equivalence invariance of formal category theory

Equipments, a special kind of double categories, have shown to be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments, and combine this together with Makkai’s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts) and Henry’s recent connection between model structures and formal languages, to show a result on the equivalence invariance of formal category theory.

Orateur: Paula Verdugo

11:10 Pause café

26 Algebras over linear infinity-operads

Using two different subcategories A and R of Omega (the category of trees), we first define linear infinity-operads as some presheaves (over A with values in chain complexes) with additional structure maps inducing a "composition up to homotopy"). We then define algebras over such an infinity-operad X as presheaves (over R with values in chain complexes) with structure maps encoding an "action up to homotopy" of X.
We will give some examples and some intuition behind these definitions.
For these operads and algebras, we define generalized bar&cobar constructions and prove they satisfy a Koszul duality.
This is a joint work with Ieke Moerdijk.

Orateur: Éric Hoffbeck

13:00 Repas à l'upsidum