To a topological space one may associate an infinity-groupoid of paths, with objects given by points, 1-morphisms given by paths, 2-morphisms given by homotopies, and so on. It is a result due to Kan and Quillen - sometimes called the homotopy hypothesis - that this construction induces an equivalence between the homotopy theory of CW-complexes and infinity-groupoids. The stratified world admits a similar construction, the infinity-category of exit paths, which associates to a sufficiently regular stratified space (such as a pseudo manifold) an infinity-category given by paths which ascend in the stratification. It was conjectured by Ayala, Francis and Rosenblyum that this construction induces an equivalence between a homotopy theory of appropriate topological stratified spaces (obtained by inverting stratified homotopy equivalences) and so called layered infinity-categories (such infinity categories in which every endomorphism is an isomorphism). In this talk, we are going to provide an affirmative answer to this conjecture. Namely, we identify a category of stratified spaces, which contains classical examples such as piecewise-linear pseudo-manifolds, such that its localization at stratified homotopy equivalences is equivalent to the homotopy theory of small layered infinity-categories, via the exit-path construction.

In this talk we consider generalisations of the Word-Splice/Contour Adjunction between Categories and Multicategories, introduced by Paul-André Melliès and Noam Zeilberger. We observe that the counit is 'almost' invertible, failing only to be full for a fairly trivial reason. This trivial reason can be circumvented by generalising the construction to polycategories. We also observe that the word-splice functor always supplies cyclic multicategories. This motivates cyclic polycategories as the most appropriate setting in which to consider this adjunction.

However there are a number of ways in which we could consider a polycategory to be cyclic. In this talk we discuss two such ways, where in the one we impose a duality condition and in the other we do not. We end by discussing some links between these cyclic polycategories and cyclic linear logic

The classical statement of the Chomsky-Schützenberger representation theorem says that any context-free language may be represented as the homomorphic image of the intersection of a Dyck language of balanced parentheses with a regular language. In the talk I will discuss a fibrational perspective on context-free grammars and finite-state automata that grew out of a long-running project with Paul-André Melliès on type refinement systems, but with a surprising twist that only emerged when we considered the C-S theorem. It turns out that underlying that theorem is a basic adjunction between categories and colored operads (= multicategories), where the right adjoint $W : Cat \to Oper$ builds a “spliced arrow operad” out of any category, and the left adjoint $C : Oper \to Cat$ sends any operad to a “contour category” whose arrows have a geometric interpretation as oriented contours of operations.

Based on joint work with Paul-André Melliès that appeared at MFPS 2022 (https://hal.science/hal-03702762), as well as a long version of that article in preparation.

Un résultat classique mais peu connu de la théorie des modèles donne une correspondance entre des structures relationnelles sur un ensemble et les sous-groupes fermés de son groupe topologique d'automorphismes. Avec la théorie des (1)-topos classifiant, on peut étendre ce théorème, remplaçant “ensemble” par “modèle d'une théorie géométrique quelconque”, et si on le veut, “groupe” par “monoïde”. Je vais expliquer ce résultat et, le temps permettant, considérer comment l'étendre aux (infini,1)-topos.

Based on a new concept of first-order generalized sketches we coined lately “Logics of Statements in Context” to provide a unified view on formalisms like Algebraic Specifications, Prolog, First-Order Logic, Ehresmann Sketches, Description Logics, Generalized Sketches à la Makkai/Diskin, Diagram Predicate Framework, Graph Conditions, and others. In the talk we present Generalized Sketches à la Makkai/Diskin as a quite natural generalization of traditional Ehresmann sketches. Generalized Sketches à la Makkai/Diskin can be defined in arbitrary categories. They built upon “atomic statements in context” and utilize sketch implications for axiomatization purposes. Going beyond atomic statements, we outline the definition of arbitrary first-order statements in arbitrary categories enabling us to enhance the expressiveness of Generalized Sketches. In analogy to first-order statements, we can also define arbitrary first-order sketch conditions generalizing thereby different kinds of “nested graph constraints and conditions”.

We intend to discuss, on the way, two essential constructions Makkai’s work on Generalized Sketches relies on: “Syntactic representation of models” and “internalization of atomic statements”.

Zoom meeting registration link : https://u-paris.zoom.us/meeting/register/tZUtdOmuqj4pGNz_ymC1VNXs6xhvCfd7Sprs

Concurrent Kleene algebra, introduced by Tony Hoare et.al. in 2011, extends Kleene algebra with a parallel composition operator. The result is a double monoid with a lax interchange law between concatenation (i.e. serial composition) and parallel composition. Its free models are series-parallel pomsets, that is, partial strings which a freely generated from the alphabet by binary serial and parallel composition.

We have recently had occasion to consider a generalisation of serial composition where events in pomsets may continue across compositions. The resulting algebraic structure is a 2-category with a form of lax tensor; for the time being it is not clear what are its free models, nor whether they even exist. Further, ultimately we is interested in languages of pomsets, and lifting 2-categories (or just categories) across the powerset functor has turned out to be interesting just in and of itself.

Joint work with Christian Johansen, Georg Struth and Krzysztof Ziemiański. References: - Posets With Interfaces as a Model for Concurrency. Information and Computation 285(B):104914 (2022). https://arxiv.org/abs/2106.10895 - Catoids and Modal Convolution Algebras. Algebra Universalis 84(10) (2023). https://link.springer.com/article/10.1007/s00012-023-00805-9

The action of a bicategory generalizes monoidal action and leads to a bicategory of compound optics. These optics are expressed using coends; they also have Tambara representations.

Double categories are a two-dimensional structure consisting of objects, two classes of morphisms (horizontal and vertical), and cells between them. A double category may be defined as an internal category in CAT, and it is called right-connected if its identity-assigning map is right adjoint to its codomain-assigning map. The intuition is that every vertical morphism in a right-connected double category has an underlying horizontal morphism. Right-connected double categories play an important role in the characterisation of algebraic weak factorisation systems, and this motivates the question: is it possible to complete a double category under the property of right-connectedness?

In this talk, I will provide an explicit characterisation of the right-connected completion of a double category, and study its properties. I will examine several examples of right-connected double categories arising from this completion, and show how particular instances lead to examples of algebraic weak factorisation systems. The main theorem of the talk will provide conditions for a double category arising from the right-connected completion to be comonadic (in a suitable sense) over the original double category.

La notion de foncteur “smothering” (étouffant en français …), introduite par Riehl et Verity, apparaît fréquemment en homotopie, par exemple le foncteur de comparaison entre la catégorie homotopique de $Ho(C^\rightarrow)$ et la catégorie de flèche $Ho(C)^\rightarrow$ pour toute quasicatégorie $C$. Une légère variation de cette notion se trouve définir la classe d'équivalence faible d'une localisation de Bousfield à droite de la structure de modèle naturelle sur Cat. Le but de l'exposé est d'illustrer l'importance de cette structure en définissant une structure de modèle induite sur la catégorie des semidérivateurs montrant que ces derniers modélisent les $(\infty,1)$-catégories.

Bien que le théorème de cohérence de MacLane pour les catégories monoïdales ait été prouvé initialement par une méthode proche de la réécriture, il présente un caractère topologique. C’est ce qui a poussé Kapranov à suggérer en 1993 une « preuve instantanée » de ce théorème basée sur l’existence d'une famille de polytopes qu’il nomme permutoassociaèdres. Dans la première partie, on formalisera cette idée, montrant que la cohérence de MacLane est une conséquence directe de la simple connexité des permutoassociaèdres. Cela suggère un lien combinatoire plus général entre cohérence n-catégorique et n-connexité de certains espaces, avatar « strict » de travaux infini-catégoriques récents de Shaul Barkan.

Dans la deuxième partie, on s'intéressera à l'instanciation du théorème général à la classe des “hypergraph polytopes” (aussi connus sous le nom de nestoèdres), dont les faces (et en particulier les sommets et les arêtes) sont décrits par des objets combinatoires appelés “constructs”. Les arêtes sont naturellement orientées à l'aide d'un ordre appelé ordre de Tamari généralisé, et lorsqu'on instancie encore davantage, à la classe des opéraèdres, on obtient un système de réécriture de termes convergent (sous-jacent au problème de cohérence des opérades catégorifiées de Dosen et Petric), ce qui permet dans ce cas d'obtenir la cohérence par des méthodes de réécriture (complétion de Squier). Les diagrammes obtenus sont légèrement différents de ceux de Dosen et Petric.

Je présenterai un travail récent en collaboration avec Dimitri Ara et Yves Lafont sur une approche des orientaux de Street proposée par Albert Burroni. Nous décrirons une monade T sur la catégorie des omega-catégories strictes construisant la suite des orientaux par itération à partir de la catégorie initiale.

https://arxiv.org/abs/2209.08022