La construction de Grothendieck est une construction fondamentale de la théorie des catégories. Dans cet exposé, j'en donnerai une généralisation dans le cadre des $\omega$-catégories faibles. À cette fin, je présenterai quelques concepts importants de la théorie des $\omega$-catégories faibles, tels que les transformations lax, les (co)limites lax, et les fibrations cartésiennes. Enfin, j'expliquerai comment ce résultat peut être utilisé pour donner des calculs explicites d'extensions de Kan lax.

n this talk, we shall introduce a calculus for a presentation of compact, orientable, connected 3-manifolds with boundary in terms of diagrams embedded in S^3 in a form akin to the standard surgery presentation of closed, orientable, connected 3-manifolds. The motivation to introduce such a presentation of manifolds is to give a completely combinatorial description of the category 3Cob, whose arrows are 3-dimensional cobordisms, which is an ongoing project. We hope this could support further investigations of faithfulness of 3-dimensional Topological Quantum Field Theories.

This is a joint work with Bojana Femić, Vladimir Grujić and Zoran Petrić.

It is well-known that the arrows of the free cartesian category C(S) over a signature S are (tuples of) terms built over S, and that cartesian functors from C(S) into Set corresponds algebras over S. A term-like structure is also available for the arrows of some free categories over S laying between the free monoidal and the free cartesian categories, and suitable functors into Rel and Par can also be interpreted as multi- and partial algebras, respectively. Finally, all these categories can be obtained as the Kleisli category of suitable monads, hence offering a tool for the “computations and monads” paradigm.

In higher category theory, model categories constitute a powerful manner to encode infinity categories. Unfortunately, it is not always possible to encode an infinity category using a model category, and when this is possible the choice of model category is far from being unique or canonical. Favoring such presentations whenever possible, it is hence natural to investigate which operations on the level of infinity categories can be performed on the model categorical level. In this talk I will describe one such result for the operation of lax limits, showing that, under suitable conditions, if a diagram of inifinity categories is modeled by a diagram of (simplicial combinatorial) model categories, then the lax limit can be performed on the level of model categories. We also obtain results concerning homotopy limits and various intermediate limits. This generalizes previous results of Lurie and of Bergner. Related results were also obtained independently by Balzin.

Je propose la notion d'un système de types hiérarchiques, de sorte que la définition soit elle même exprimée de manière interne au système qu'elle définit. L'idée est qu'un tel système fournit un cadre au sein duquel on peut raisonner, tout en possédant sa propre logique. On pourra établir des constructions formelles en se plaçant dans un système quelconque et en déduire des résultats généraux. On définira notamment la notion de oméga type dans un système. On verra que le cas non trivial minimal émane naturellement de la logique booléenne habituelle et fournit la hiérarchie des n-catégories. On remarquera la nature intrinsèquement homotopique du cadre ainsi obtenu, si bien que les oméga types de ce système peuvent naturellement s'interpréter comme des espaces. J'aimerai aussi revenir sur quelques axiomes de la théorie homotopique des types et de la théorie des ensembles.

In the setting of homotopy type theory, each type can be interpreted as a space. Moreover, given an element of a type, i.e. a point in the corresponding space, one can define another type which encodes the space of loops based at this point. In particular, when the type we started with is a groupoid, this loop space is always a group. Conversely, to every group we can associate a type (more precisely, a pointed connected groupoid) whose loop space is this group: this operation is called delooping. The generic procedures for constructing such deloopings of groups (based on torsors, or on descriptions of Eilenberg-MacLane spaces as higher inductive types) are unfortunately equipped with elimination principles which do not directly allow eliminating to arbitrary types, and are thus difficult to work with in practice. Here, we construct deloopings of the cyclic groups Z_m which are cellular, and thus do not suffer from this shortcoming. In order to do so, we provide type-theoretic implementations of lens spaces, which constitute an important family of spaces in algebraic topology. In some sense, this work generalizes the construction of the real projective space by Buchholz and Rijke in their LICS'17 paper, which handles the case m=2, although the general setting requires more involved tools. Finally, we use this construction to also provide cellular descriptions of dihedral groups, and explain how we can hope to use those to compute the cohomology and higher actions of such groups.

L'idée est de présenter quelques formalisations des opétopes (entre 3 et 4…) avec lesquelles j'ai pu travailler durant mon stage. Et notamment d'introduire des définitions originales de Pierre-Louis Curien et moi. Je ne pense pas parcourir la preuve de l'équivalence, éventuellement donner des idées vagues de ce qu'il faut faire pour les relier.

Si le temps le permet, je pourrait aussi introduire une définition pour les opétopes (non nécessairement positifs cette fois) à laquelle j'ai réfléchi dernièrement, et dont j'ai démontré l'équivalence avec les “Zoom complexes” du papier “Polynomial functors and opetopes” par Kock, Joyal, Batanin et Mascari.