
The seminar is on hold following the announcement of the second period of confinement. Further information will be available as the situation evolves.

15 et 22 octobre 2020 à 10h30, Salle II
Victor Iwaniack
We will study a categorification of automata theory where automata and languages are viewed as functors. We will then see how we can build classical constructions such as the initial, final and minimal automata using category theoretic notions like factorization systems and Kan extensions. 
24 septembre 2020, 1 octobre 2020, 8 octobre 2020 à 10h30, Salle I
Jérémie Marquès
We will see how some firstorder theories containing a linear order with endpoints correspond to some monoïds in the category of Stone spaces, using doctrines (an algebraization of firstorder logic) and their duals polyadic algebras. This can be used to associate a firstorder theory to some boolean algebras of regular languages. 
12 mars 2020 à 10h30, Salle de conférence
PierreLouis Curien
Opetopes are shapes, which can be formalised as trees whose nodes are themselves trees whose nodes etc., that were introduced by Baez and Dolan for an approach to a definition of higher categories. But opetopes are difficult combinatorial objects. In joint work with Cédric Ho Thanh and Samuel Mimram, we have explored various ways to describe opetopes formally. One of these ways is to make a systematic use of higher addresses, which are words of words of words etc., based on the remark that nodes in trees are identified by their parth to the root. I shall intoduce the corresponding encoding/presentation of opetopes, Then I’ll show on top of it a syntactic presentation of the category of opetopic sets, and of the axioms of higher categories as described by Finster. In joint work just started with Amar Hadzihasanović and Samuel Mimram, we aim at developing in parallel two approaches to categories of opetopes: one where degeneracies (used to define units when it comes to higher categorical structure) come from objects (like in the category of opetopes described in the work of Curien  Ho Thanh  Mimram), and the other where it comes from the morphisms (objects being then limited to positive opetopes, while morphisms are not only face inclusions but can also feature contractions of globes). I’ll show how to adapt the syntactic machinery to this other setting. 
6 fevrier 2020 à 10h30, Salle de conférence
Brett McLean
Two important classes of wordtoword functions related to finitestate transducers are the rational functions and the regular functions. The rational functions are those partial functions realised by oneway nondeterministic transducers. The regular functions are those realised by twoway nondeterministic, or equivalently deterministic, transducers. Both these classes are closed under the following operations that have been studied in the literature on partial functions: composition, antidomain, range, and preferential union (also known as override). I will give an overview of a categorical duality having on one side the (isomorphs of) algebras of partial functions equipped with these four operations, and on the other side certain topological partial algebras. This generalises a duality due to Mark Lawson between a certain category of inverse semigroups and a certain category of topological groupoids. At the end I may briefly speculate about applications of the duality to the study of rational and regular functions, and solicit suggestions on this matter. 
23 janvier 2020 à 14h00, Salle I
Luck Darnière

5 décembre 2019 à 10h30, Salle de conférence
Maria Manuel Clementino
Having as starting point Escardó's result on injectivity for lax idempotent, or KockZoberlein, monads [5, 6], we will show how filter monads in T0spaces can be used to study fibrewise injectivity [2]. This will be the starting point to introduce KZreflective factorisation systems [3, 4], generalising the reflective factorization systems of CassidyHébertKelly [1]. Examples of such factorization systems include interesting classes of continuous maps, like fibrewise versions of continuous lattices and continuous Scott domains, or of stable compactness and sobriety. We will conclude listing some open problems.
References:
[1] C. Cassidy, M. Hébert, G.M. Kelly, Reflective subcategories, localizations and factorization systems, J. Austr. Math. Soc. Ser. A 38 (1985) 287329.
[2] F. Cagliari, M.M. Clementino, S. Mantovani, Fibrewise injectivity and KockZoberlein monads, J. Pure Appl. Algebra 216 (2012) 24112424.
[3] M.M. Clementino, I. López Franco, Lax orthogonal factorisatio systems, Adv. Math. 302 (2016) 458528.
[4] M.M. Clementino, I. López Franco, Lax orthogonal factorisation in ordered structures, arXiv 1702.02602.
[5] M. Escardó, Properly injective spaces and function spaces, Topology Appl. 89 (1998) 75120.
[5] M. Escardó, R. Flagg, Semantic domains, injective spaces and monads, Electr. Notes in Theor. Comp. Science 20 (1999) 15. 
28 novembre 2019 à 10h30, Salle de conférence
Carlos Simpson
This is to report on our ongoing projects with Allouch, GhannoumFussnerJakl, BalzinShminke, Alfaya, ..., to investigate the world of finite semigroups and finite categories. We'll first discuss the analysis of a semigroup by twosided ideals, and irreducible objects. Then, the relationship between these and 2element categories. Then we'll look at various attempts to use machine learning to look at the landscape. 
7 novembre 2019 à 10h30, Salle de conférence
Tomáš Jakl
A systematic theory of structural limits for finite models has been developed by Nešetřil and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises  via StonePriestley duality and the notion of types from model theory  by enriching the expressive power of firstorder logic with certain "probabilistic operators". We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction.
The consequences are twofold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the dualitytheoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively. 
17 octobre 2019 à 2h00, Salle I
Joan Millès
Les algèbres courbées ne vérifient pas l’identité fondamentale : d^2 = 0. Parler d’homologie pour ces algèbres n’a donc pas de sens a priori. Elles apparaissent cependant dans des contextes (en géométrie par exemple) où un analogue d’une théorie de l’homotopie est requise. La dualité de Koszul est au coeur de la théorie de l’homotopie des algèbres. Pour l’étendre aux algèbres courbées, nous nous plaçons dans un contexte filtré et complet qui amène des problèmes catégoriques pour lesquels nous présenterons des solutions.  3 octobre 2019 à 10h30, Salle de conférence
Marco Abbadini
Dualities have been proved to be of great interest. One way to obtain a duality is to first establish a dual adjunction between two categories X and A (i.e., an adjunction between X and the opposite of A) and then restrict such dual adjunction to those objects at which the unit is an isomorphism. For example, Stone duality can be obtained as the restriction of a dual adjunction between topological spaces and Boolean algebras. This motivates our interest in dual adjunctions, and the aim of this talk is to discuss “how to establish a dual adjunction with as little effort as possible”. To establish a dual adjunction, one should verify some propertiessuch as functoriality of the functors, naturality of the units and triangular identitieswhose proof is often tedious. We outline a general context in which we prove the aforementioned properties once for all; this context encompasses many known dualities, such as Stone, Priestley, Esakia, JónssonTarski, Gelfand for C*algebras. In this general setting, we also discuss a condition which is sufficient (but not necessary) to settle the remaining properties; this condition is inspired by a known condition of existence of initial lifts, and applies also to nonnatural dual adjunctions. 
24 septembre 2019 à 10h30, Salle II
Frederik Lauridsen
Recently Ciabattoni, Galatos, and Terui have introduced a new notion of completion of residuated lattices which they call the hyperMacNeille completion. This type of completion plays a central role in their work on establishing the admissibility of the cutrule in certain types of hypersequent calculi for substructural logics. In this talk we will focus on describing the hyperMacNeille completion of Heyting algebras. We show that the hyperMacNeille completion and the MacNeille completion coincide for socalled centrally supplemented Heyting algebras. That is, Heyting algebras with orderduals being Stone lattices. In fact, we show that any Heyting algebra A has a centrally supplemented extension S(A) such that the hyperMacNeille completion of A is isomorphic to the MacNeille completion of S(A). These centrally supplemented extensions turn out to be the orderduals of the 1Stone extensions originally introduced by Davey in 1972. Time permitting, we will also discuss sheaf representations of centrally supplemented Heyting algebras and the problem of describing the MacNeille completions of lattices of global sections of a sheave of lattices over a Boolean space. This is joint work with John Harding of New Mexico State University.
References:
[1] A. Ciabattoni, N. Galatos, and K. Terui, Algebraic proof theory: Hypersequents and hypercompletions. Ann. Pure Appl. Logic, 186(3) 693737 (2017).
[2] G. D. Crown, J. Harding, and M. F. Janowitz, Boolean products of lattices. Order 13(2) 175205 (1996).
[3] B. A. Davey mStone lattices, Canadian J. Math., 24(6) 10271032 (1972).
[4] J. Harding, Completions of orthomodular lattices II. Order 10(3) 283294 (1993).