## Séminaire du 12 octobre 2018 à Lille

ATTENTION : Ce SIC remplace celui initialement annoncé pour le 18 mars 2018, reporté à cause de la “grève perlé” SNCF du printemps dernier.

Lieu : Université de Lille, Laboratoire de Mathématiques, Bâtiment M2, Salle de réunion.

Organisation :

Ivo Dell’Ambrogio <ivo.dell-ambrogio@univ-lille.fr>
Isar Stubbe <isar.stubbe@lmpa.univ-littoral.fr>
Alexis Virelizier <alexis.virelizier@univ-lille.fr>

Ce SIC est soutenu financièrement par la Fédération de Recherche Mathématique du Nord-Pas-de-Calais et par le Labex CEMPI.

Participation :

La participation au SIC est gratuite. Pour des raisons pratiques, on demande aux participants de s’inscrire. Pour cela, un simple mail à un des organisateurs suffit. Le repas de midi sera offert aux participants inscrits avant le 1 octobre.
Lors de l’inscription, veuillez nous communiquer votre choix entre les deux plats suivants :
1) suprême de poulet sauce forestière (poultry meat),
2) dos colin crème citronné (fish).
Le repas comprendra aussi un buffet d’hors d’oeuvres et un buffet de desserts.

Programme :

10h15 : Accueil avec café
10h50 – 11h40 : Jacques Darné (Grenoble)
11h50 – 12h40 : Tom Hirschowitz (Chambéry)
Repas à l’Ascotel
14h00 – 14h50 : Simon Henry (Brno)
15h00 – 15h50 : Paolo Saracco (Bruxelles)
Pause café
16h10 – 17h00 : Idriss Tchoffo Nguefeu (Louvain-la-Neuve)

Titres des exposés :

Jacques Darné : Actions in the category of N-series (résumé).
Tom Hirschowitz : Foncteurs polynomiaux et carrés exacts en sémantique des jeux.
Simon Henry : La conjecture de strictification de Simpson (résumé).
Paolo Saracco : Tannaka-Kreĭn reconstruction and coquasi-bialgebras with preantipode (résumé).
Idriss Tchoffo Nguefeu : Connecteurs et groupoïdes internes dans les categories de Goursat (résumé).

Participants (ordre d’inscription) :

Ivo Dell’Ambrogio (Lille)
Isar Stubbe (Calais)
Alexis Virelizier (Lille)
Jacques Darné (Grenoble)
Simon Henry (Brno)
Tom Hirschowitz (Chambéry)
Paolo Saracco (Bruxelles)
Idriss Tchoffo Nguefeu (Louvain-la-Neuve)
Andrée Ehresmann (Amiens)
Giulio Calimici (Lille)
Ivan Bartulovic (Lille)
Antoine Touzé (Lille)
James Huglo (Lille)
Jun Maillard (Lille)
Dominique Bourn (Calais)
John Robert (Bruxelles)
Daniel Tanré (Lille)
Huafeng Zhang (Lille)
Aurélien Djament (Lille)
Benoit Fresse (Lille)
Tim van der Linden (Louvain-la-Neuve)
Amine Laaroussi (Lille)
Hongyi Chu (Lille)

Informations utiles :

Pour venir à Lille :
Plan d’accès
Instructions détaillées

Pour dormir à Lille :
Office de Tourisme de Lille

Résumés :

Jacques Darné : Actions in the category of N-series

N-series are nicely-behaved filtrations on groups, linked to their nilpotent structure. We will introduce a category of N-series. This category is homological, and the general definition of actions in such categories apply. It admits universal actions, which are linked to generalized Johnson morphisms. We will describe how this language allows us to get an interesting point of view on some classical constructions, and to get some new results about them. (Retour ↩)

Simon Henry : La conjecture de strictification de Simpson

In ’91, Kapranov and Voevodsky erroneously claimed to prove that every weak infinity groupoid (homotopy type) is equivalent to one in which all the composition operations (except inverses) are strict, i.e. a strict infinity category with weak inverse. This is now known to be false, but Carlos Simpson conjectured in ’98 that any weak infinity groupoid is equivalent to one were all composition operations are strict, except units and inverses, i.e. a “non-unital strict infinity category” with weak units and weak inverses. He also mentioned that Kapranov and Voevodsky’s paper might contains the proof of this claim. In this talk I will present very recent progress on this conjecture, including a first proof of a form of the conjecture in arbitrary dimension, and which indeed use the same ideas as Kapranov and Voevodsky’s original proof. (Retour ↩)

Paolo Saracco : Tannaka-Kreĭn reconstruction and coquasi-bialgebras with preantipode

Via the Tannaka-Kreĭn formalism it is possible to construct a Hopf algebra from a rigid monoidal category endowed with a monoidal “fiber” functor to finitely-generated and projective k-modules. In particular, this allows us to characterize Hopf algebras over a field as those bialgebras whose category of finite-dimensional corepresentations is rigid. In this talk, we report on an analogue reconstruction theorem and we show how a rigid monoidal category with a quasi-monoidal “fiber” functor gives rise to a coquasi-bialgebra with preantipode (as introduced by Ardizzoni and Pavarin). When k is a field, this allows us to characterize coquasi-bialgebras with preantipode and coquasi-Hopf algebras in terms of the rigidity of their categories of finite-dimensional comodules.

As an application, we will endow the finite dual coalgebra of a quasi-bialgebra with preantipode with the structure of a coquasi-bialgebra with preantipode. (Retour ↩)

Idriss Tchoffo Nguefeu : Connectors and internal categories in Goursat categories

(Joint work with Marino Gran and Diana Rodelo.) Given two equivalence relations $(R, r_1, r_2)$ and $(S, s_1, s_2)$ on the same object $X$, we denote by $R\times_X S$ the pullback
$\begin{matrix} R\times_X S & \rightarrow & S \\ \downarrow & & \downarrow s_1 \\ R & \stackrel{r_2}{\rightarrow} & X \end{matrix}$
In a finitely complete category $\mathcal{C}$ a connector [2] between $R$ and $S$ is an arrow $p\colon R\times_X S \rightarrow X$ such that (internally)
$\bullet xSp(x,y,z)Rz$;
$\bullet p(x,x,y) = y$ and $p(x,y,y) = x$,
$\bullet p(x,y,p(z,u,v))=p(p(x,y,z),u,v)$,
whenever each term is defined. One of the main interests of the notion of connector is that it allows one to study centrality of equivalence relations in any Mal’tsev category even without defining the commutator of equivalence relations. Also in any Goursat category [1] a connector between $R$ and $S$ is unique when it exists, since the Shifting Lemma holds in any Goursat category [3]. In this talk, we prove that connectors are stable under quotients in any regular Goursat category. As a consequence, the category $Conn(\mathcal{C})$ of connectors in $\mathcal{C}$ is a Goursat category whenever $\mathcal{C}$ is. This implies that Goursat categories can be characterised in terms of a simple property of internal groupoids [4]. We finally prove that regular Mal’tsev and Goursat categories can be characterised through stronger variations of the Shifting Lemma [5].
References
[1] A. Carboni, G.M. Kelly, M.C. Pedicchio, Some remarks on Mal’tsev and Goursat categories, Appl. Categ. Structures 1 (1993), no. 4, 385-421.
[2] D. Bourn and M. Gran, Centrality and connectors in Maltsev categories, Algebra Universalis 48 (2002), no. 3, 309-331.
[3] D. Bourn and M. Gran, Categorical Aspects of Modularity, Galois Theory, Hopf Algebras and Semiabelian Categories, Fields Instit. Commun. 43, A.M.S., Providence RI (2004) 77-100.
[4] M. Gran, D. Rodelo, I. Tchoffo Nguefeu, Some remarks on connectors and groupoids in Goursat categories, Logical Methods in Computer Science (2017), vol.13(3:14), 1-12.
[5] M. Gran, D. Rodelo, I. Tchoffo Nguefeu, Variations of the Shifting Lemma and Goursat categories, preprint, http://arxiv.org/abs/1809.10408 (2018). (Retour ↩)