- 29 mai 2019 à 10h30, Salle II
Jorge Almeida [Slides]
By a profinite congruence we mean a congruence on a profinite algebra whose corresponding quotient algebra is profinite. Profinite congruences play a role for instance in profinite presentations. In that context, one is interested in determining the profinite congruence on a relatively free profinite algebra generated by the relators, a congruence that is hard to describe. In contrast, it is easy to describe in a somewhat (transfinitely) constructive manner the closed congruence generated by a given binary relation on a profinite algebra. We show that whether a closed congruence on a profinite algebra is profinite is a purely topological property, namely, the corresponding quotient algebra being 0-dimensional. We use this result and a representation of absolutely free Stone topological algebras to deduce that every finitely generated Stone topological algebra of finite signature is profinite.
(Joint work with O. Klíma, from Masaryk University, Brno, Czech Republic)
- 20 - 24 mai 2019, Salle de Réunion Fizeau
Friedrich Wehrung, Course supplements: [1] [2] [3]
For categories A and B, the determination of the range of a given functor F: A\to B gives often rise to seemingly intractable problems, even at the most basic level — that is, does every object of B belong to the range of F (up to isomorphism of course)? Similar, apparently stronger questions can be stated for arrows in B, or, more generally, for commutative diagrams in B. It turns out that due to a 2011 construction of the author with Pierre Gillibert, that we called the condensate construction, all those questions are, under fairly general conditions, equivalent. For instance, representing an arrow is « not really » harder than representing an object. However, this equivalence comes with a cost: going from a diagram counterexample to an object counterexample (outside the range of F) requires a cardinality jump, the amplitude of which depends of a natural number called the Kuratowski index (often equal to the order-dimension) of the shape of the diagram in question.
Recent improvements of the condensate construction made it possible to prove stronger negative results, stating that the range of the functor F is not even closed under elementary equivalence with respect to infinitary languages of the form L_{\infty,\lambda}. Such negative results are inferred from the existence of a (necessarily non-commutative) diagram D in A such that F(D) is not F(X) for any commutative diagram X in A. For example, the long-standing problem of the characterization of the spectra of all Abelian lattice-ordered groups finds there a negative solution: namely, the class of all Stone duals of such spectra (which are special kinds of distributive lattices with zero) is not closed under L_{\infty,\lambda}-elementary equivalence for any infinite cardinal \lambda; in particular, it is not the class of all models of any class of L_{\infty,\lambda} sentences.
Schedule:- Monday, May 20: 15h - 16h
- Tuesday, May 21: 15h - 16h
- Wednesday, May 22: 15h - 16h
- Thursday, May 23: 11h - 12h
- Friday, May 24: 11h - 12h
- 16 mai 2019 à 10h30, Salle II
Jean-Éric Pin
A Pervin space is a set equipped with a bounded lattice of subsets. This notion allows one to give a concrete description of the dual space of this distributive lattice: it is the completion of the Pervin space.
The general question of extending a structure from sets to hypersets, is well-studied for ordered sets, topological spaces and quasi-uniform spaces, leading to divergent terminology. In this lecture, we address the case of Pervin spaces. This is an ongoing joint work with Mário Branco.
- 2 mai 2019 à 16h00, Salle I
Tom Hirshowitz
Structural operational semantics is a standard method for specifying the syntax and dynamics of programming languages. We propose a categorical framework for structural operational semantics, in which we prove that under suitable hypotheses bisimilarity is a congruence. We then refine the framework to prove soundness of bisimulation up to context, an efficient method for reducing the size of bisimulation relations. Finally, we demonstrate the flexibility of our approach by reproving known results in three variants of the pi-calculus.
- 25 avril 2019 à 10h30, Salle II
Joachim Kock
A classical theorem of Hopkins, Neeman and Thomason can be stated in the following conceptual way. For R a commutative ring, the compactly generated localising subcategories of the derived category D(R) form a coherent frame, Hochster dual to the Zariski frame (the frame of radical ideals in R). I'll explain the statement, contrasting its original formulation with the above formulation, which belongs to the setting of point-free topology, and sketch the proof, which exploits cellularisation techniques. Next I'll explain how also the Zariski frame itself can be realised inside D(R). Finally I'll comment on the global case, Thomason's theorem for coherent schemes (i.e. quasi-separated and quasi-compact), whose proof in this approach is related to recent developments in constructive algebraic geometry. This is joint work with Wolfgang Pitsch (TAMS 2017).
- 4 avril 2019 à 10h30, Salle II
Najwa Ghannoum
Category theory studies mathematical structures by abstracting the concept of a map between two such structures. A category is, in particular, a directed graph whose nodes are called objects, and whose labeled edges are called arrows.
In this talk, we will introduce the concept of finite category, and some of its important properties. The main idea is to associate to any fixed matrix (whose entries belong to N) a family of finite categories, and to study how the coefficients of the matrix determine the elements of this family. In particular, we present the set of finite categories corresponding to the square matrix of order n, with every coefficient equal to 2 or 3.
We will also present some specific matrices with small entries, and compute the exact number of finite categories associated to these matrices. Numerical data from associated semigroups sheds light on the structure of finite categories and suggests some combinatorial results. Then, we will introduce how a program can compute the number of categories of small sizes.
References:
[1] S. Allouch, C. Simpson, Classification des Matrices Associées aux Catégories Finies, Cahiers de topologie et de géométrie différentielle catégoriques 55 (2014), 205-240.
[2] S. Allouch, C. Simpson, Classification of Categories with Matrices of Coefficient 2 and order n, Communications in Algebra 46 (2018), no. 7, 3079-3091.
[3] C. Berger, T. Leinster, The Euler Characteristic of a Category as the Sum of a Divergent Series. Homol., Homotopy Appl. (2008), 10:41-51.
- 28 mars 2019 à 10h30, Salle II
Clemens Berger
Distributive lattices are a natural generalisation of Boolean lattices insofar as they satisfy all relations that hold in a Boolean lattice, except that their elements do not necessarily have complements. Another interesting generalisation of Boolean lattices consists of so-called orthocomplemented lattices in which all elements have complements, but the distributivity laws do not necessarily hold. These structures have some importance in quantum physics and quantum logic.
In this talk I will discuss a far reaching generalisation, introduced in the mid 1990's by Bennett-Foulis and Chovanec-Kopka under different names (effect algebra resp. difference poset). A difference poset is a partially ordered set with 0,1 in which a difference y-x is defined as soon as x<=y and some natural axioms are satisfied. In particular, every element possesses a complement 1-x, and there is a natural notion of orthogonality.
Recently, Geiza Jenca proved that difference posets are the algebras for a monad on bounded posets, which has been introduced by Kalmbach in the 1970's! This monad takes linear orders to Boolean lattices, and takes thus an arbitrary bounded poset to an "amalgam'" of Boolean lattices. We will discuss some aspects of this monad and state some conjectures.
- 20 - 24 mai 2019, Salle de Réunion Fizeau