
21 mars 2019 à 10h30, Salle II
Clemens Berger
Since the seminal work by Eilenberg on singular homology, simplicial structures pervade algebraic topology. In the late 1950's, Dold and Kan discovered a remarkable categorical equivalence between simplicial abelian groups and ordinary chain complexes. As a byproduct they got explicit simplicial models of socalled EilenbergMacLane spaces which are omnipresent in algebraic topology.
In this talk (based on joint work with Christophe Cazanave and Ingo Waschkies) I will outline an abstract proof of the DoldKan correspondence which applies to many other situations as well. The key of our proof is the construction of idempotent endomorphisms in the simplex category which satisfy twosided Schützenberger relations xyx=xy=yxy. These idempotents generate monoids known in literature as Catalan monoids. The fact that these Catalan monoids have a certain combinatorial property (Jtriviality) is coresponsible for the DoldKan correspondence.

7 mars 2019 à 10h30, Salle II
JeanFrançois Mascari
A quantale is a complete lattice with an associative binary operation preserving sups in both variables. We survey properties of quantic nuclei and the representation theorem for quantales. Connections with algebraic structures used in Linear Logic and Substructural Logics will be presented. Categories enriched over quantales are of interest in Duality Theory. Connections with C. Simpson's approximate categorical structures will be considered.

14 février 2019 à 10h30, Salle II
Wesley Fussner
Substructural logics are systems of reasoning that place more stringent requirements on deduction than classical logic. Whereas classical logic concerns itself with inferences about the truth of propositions, substructural logics describe deductions applying to stricter notions, such as constructability or relevance. Such deductive systems are numerous and diverse, and arise not just in mathematics, but also in computer science, philosophy, and linguistics. Nevertheless, they admit a unified algebraic treatment through a class of latticeordered monoids called residuated lattices. These play a role in substructural logics analogous to the role that Boolean algebras play in classical logic.
Here we give a brief survey of substructural logics and residuated lattices, and discuss dualitytheoretic techniques for their study. We focus in particular on the case study of Sugihara monoids, a class of idempotent residuated lattices that admits a simple topological duality casting considerable light on its algebraic structure. This topological duality is obtained as a restriction of a natural duality for an appropriatelychosen reduct of Sugihara monoids.

7 février 2019 à 10h30, Salle II
Wesley Fussner
Substructural logics are systems of reasoning that place more stringent requirements on deduction than classical logic. Whereas classical logic concerns itself with inferences about the truth of propositions, substructural logics describe deductions applying to stricter notions, such as constructability or relevance. Such deductive systems are numerous and diverse, and arise not just in mathematics, but also in computer science, philosophy, and linguistics. Nevertheless, they admit a unified algebraic treatment through a class of latticeordered monoids called residuated lattices. These play a role in substructural logics analogous to the role that Boolean algebras play in classical logic.
Here we give a brief survey of substructural logics and residuated lattices, and discuss dualitytheoretic techniques for their study. We focus in particular on the case study of Sugihara monoids, a class of idempotent residuated lattices that admits a simple topological duality casting considerable light on its algebraic structure. This topological duality is obtained as a restriction of a natural duality for an appropriatelychosen reduct of Sugihara monoids.

31 janvier 2019 à 10h30, Salle II
Irène Guessarian [Slides]
Looking at some monoids and (semi)rings (natural numbers, integers and padic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave like the operations of the algebra. The first way is to preserve congruences or stable preorders. The second way is to demand that preimages of recognizable sets belong to the lattice or the Boolean algebra generated by the preimages of recognizable sets by "derived unary operations" of the algebra (such as translations, quotients, ...).
(joint work with Patrick Cegielski and Serge Grigorieff)

24 janvier 2019 à 10h30, Salle II
Christian Retoré [Slides]
Let us call a sheaf model a (pre)sheaf of classical models (the ones of model theory). Given an open U of a (pre)topology, the truth of a formula F at U depends on the truth of the subformulas of F at subopens of U and is defined by the KripkeJoyal forcing.
We shall first present KripkeJoyal forcing introduced by Joyal. As the name suggests, it resembles Kripke models and Cohen forcing. We then show that every first order formula provable in intuitionistic logic is true in every sheaf model — soundness of this interpretation. We shall give an example of a formula F of ring theory that is classically provable but false in the sheaf of rings of continuous functions from R to R — hence F is not provable in intuitionistic logic. Finally, we shall give a direct proof, using a canonical sheaf model, of a result of the 80s, namely the completeness of intuitionistic logic with respect to sheaf models: a first order formula that is true in every sheaf model is provable in first order intuitionistic logic (the converse of soundness above). As a conclusion, we shall discuss future variants of this result according to the nature of the presheaf (separated presheaf, sheaf) and to the nature of the topological space (pretopology, topology, standard topological space). We shall end up with a question: does such results hold for first order modal logic (S4, which resembles intuitionistic logic).
This work includes contributions by Jacques van de Wiele (U. Paris 7 Diderot), Ivano Ciardelli (Ludwig Maximilian U., München) and David Theret (U. Montpellier).

17 janvier 2019 à 10h30, Salle II
Tomáš Jakl
Recently in finitely model theory and in combinatorics a number of people started studying limits of finite structures or graphs. Understanding such limits is important for the general study of large networks arising in biology, economy, physics, social sciences, etc.
In this talk we will give an overview of some of the basic facts in the study of limits of finite structures. In particular we focus on recent approach of Jaroslav Nešetřil and Patrice Ossona de Mendez [1,2]. The authors embed finite structures in the space of all measures and then compute the limits in this space.
The main representation theorem by Nešetřil and Ossona de Mendez uses tools from functional analysis. We give an alternative proof which uses canonical extensions known from lattice theory instead.
References
[1] Jaroslav Nešetřil, Patrice Ossona de Mendez. A Model Theory Approach to Structural Limits. Commentationes Mathematicae Universitatis Carolinae, 53.4 (2012), 581603 (2012).
[2] Jaroslav Nešetřil, Patrice Ossona de Mendez. A Unified Approach to Structural Limits, and Limits of Graphs with Bounded TreeDepth. arXiv preprint arXiv:1303.6471 (2013).
[3] Mai Gehrke. Canonical Extensions, Esakia Spaces, and Universal Models. Leo Esakia on duality in modal and intuitionistic logics. Springer, Dordrecht, 941 (2014).

10 janvier 2019 à 10h30, Salle II
Tomáš Jakl
Recently in finitely model theory and in combinatorics a number of people started studying limits of finite structures or graphs. Understanding such limits is important for the general study of large networks arising in biology, economy, physics, social sciences, etc.
In this talk we will give an overview of some of the basic facts in the study of limits of finite structures. In particular we focus on recent approach of Jaroslav Nešetřil and Patrice Ossona de Mendez [1,2]. The authors embed finite structures in the space of all measures and then compute the limits in this space.
The main representation theorem by Nešetřil and Ossona de Mendez uses tools from functional analysis. We give an alternative proof which uses canonical extensions known from lattice theory instead.
References
[1] Jaroslav Nešetřil, Patrice Ossona de Mendez. A Model Theory Approach to Structural Limits. Commentationes Mathematicae Universitatis Carolinae, 53.4 (2012), 581603 (2012).
[2] Jaroslav Nešetřil, Patrice Ossona de Mendez. A Unified Approach to Structural Limits, and Limits of Graphs with Bounded TreeDepth. arXiv preprint arXiv:1303.6471 (2013).
[3] Mai Gehrke. Canonical Extensions, Esakia Spaces, and Universal Models. Leo Esakia on duality in modal and intuitionistic logics. Springer, Dordrecht, 941 (2014).

29 novembre 2018 à 10h30, Salle de réunion Fizeau
Tomáš Jakl
Recently in finitely model theory and in combinatorics a number of people started studying limits of finite structures or graphs. Understanding such limits is important for the general study of large networks arising in biology, economy, physics, social sciences, etc.
In this talk we will give an overview of some of the basic facts in the study of limits of finite structures. In particular we focus on recent approach of Jaroslav Nešetřil and Patrice Ossona de Mendez [1,2]. The authors embed finite structures in the space of all measures and then compute the limits in this space.
The main representation theorem by Nešetřil and Ossona de Mendez uses tools from functional analysis. We give an alternative proof which uses canonical extensions known from lattice theory instead.
References
[1] Jaroslav Nešetřil, Patrice Ossona de Mendez. A Model Theory Approach to Structural Limits. Commentationes Mathematicae Universitatis Carolinae, 53.4 (2012), 581603 (2012).
[2] Jaroslav Nešetřil, Patrice Ossona de Mendez. A Unified Approach to Structural Limits, and Limits of Graphs with Bounded TreeDepth. arXiv preprint arXiv:1303.6471 (2013).
[3] Mai Gehrke. Canonical Extensions, Esakia Spaces, and Universal Models. Leo Esakia on duality in modal and intuitionistic logics. Springer, Dordrecht, 941 (2014).

22 novembre 2018 à 10h30, Salle de réunion Fizeau
Angel Toledo
Since their introduction in the 60's, torsion theories have been used in the study of rings through associated abelian or triangulated categories. In this talk I will give a general overview of the theory including some basic constructions and examples.
I will then present a classical result which establishes an homeomorphism between the Zariski spectrum associated to a commutative ring R, and a subspace of the lattice Rtors of the torsion theories associated to R.
The triangulated and noncommutative cases will be then discussed
References
[1] Stenström, B. (2012). Rings of quotients: An introduction to methods of ring theory.
[2] Golan, J. (1974). Topologies on the torsiontheoretic spectrum of a noncommutative ring. Pacific Journal of Mathematics, 51(2), 439450.
[3] Beligiannis, A., & Reiten, I. (2007). Homological and homotopical aspects of torsion theories. American Mathematical Soc..
[4] Van Oystaeyen, F. (2007). Virtual topology and functor geometry. Chapman and Hall/CRC.

15 novembre 2018 à 10h30, 1er étage Fizeau
Almudena Colacito [Slides]
The spectral space of a latticeordered group is defined as the set of its prime convex sublattice subgroups with the hullkernel (or spectral, or Stone, or Zariski) topology. This notion of spectral space was initially introduced for Abelian latticeordered groups by Klaus Keimel in his doctoral dissertation, following on from the success of scheme theory in algebraic geometry. For a latticeordered group, its spectrum can also be thought of as the Stone dual of the distributive lattice (with 0) of its principal convex sublattice subgroups and, as such, it is a generalized spectral space. Here, the term ‘generalized spectral’ refers to a space that is not necessarily compact but satisfies all other properties in Hochster’s definition of spectral space. Moreover, spectra of latticeordered groups are completely normal, in the sense that the specialization order associated to the topology is a root system.
In 2004, Adam Sikora’s “Topology on the spaces of orderings of groups” pioneered a different perspective on the study of the interplay between topology and ordered groups, that has led to applications to both orderable groups and algebraic topology. The basic construction in Sikora’s paper is the definition of a topology on the set of right orders on a given right orderable group. The topology is then proved compact, Hausdorff, and zerodimensional.
The theory of latticeordered groups and the theory of right orderable groups have been proved to be deeply related, and examples of this interdependence can be found almost everywhere in the literature of either field. E.g., every latticeordered group is right orderable as a group, and its lattice order can be obtained as the intersection of some of its right orders. For this reason, the question whether a relation can be found between the topological space of right orders of a right orderable group, and the spectrum of some latticeordered group arises naturally.
In this talk, I will present joint work with Vincenzo Marra, where we provide a positive answer to this question. In order to give a satisfying result that intrinsically relates the two topological spaces, we employ a fully general and natural construction involving all the varieties of latticeordered groups. So as to talk about the whole spectrum on the latticeordered group side, it is necessary to consider the broader notion of (right) preorder on the group side. The resulting correspondence entails a few immediate consequences, including the possibility of representing certain latticeordered groups via the space of (right) orders on a group. If time allows, I will discuss some applications, and possibilities for further research.

8 novembre 2018 à 10h30, Salle de conférence
Axel Osmond
Some of the most prominent dualities between spaces and algebras, such as Stonelike dualities for ordered structures or Grothendieck duality between schemes and rings, may be seen as instantiations of a common construction, namely a spectral construction. It consists primarily of a contravariant adjunction given by:
 A spectrum functor assigning to any model of an algebraic theory a certain site equipped with a structural sheaf. The site carries a topology implicitly encoded in the algebraic structure of each model and its place inside the ambient category. Under some conditions it is just a space, while in others a sheaf is needed. The sheaf encodes geometric and local behavior, that is, all the nondualizable information that remains.
 A left adjoint reconstructing each algebraic structure from the geometric information attached to the associated structured space, or in some cases, directly from its topology.
In the most general setting, such a construction takes place in a category of models of an essentially algebraic theory, that is, a locally finitely presentable category. It is produced by the presence of an orthogonality/lifting structure, whose restriction over finitely presented objects mimics a topological behavior, with notions of points, local objects, etale opens and covers. Dually, since finitely presented objects mirror the syntactic category of the underlying theory, those topological data define some extension of this theory by axioms of geometric logic.
The topological and geometric data produced through this process can be gathered into the so called spectrum associated to setvalued models of the theory. However this construction does not only concern setvalued models. The induced topology defines a certain subtopos of the classifying topos of the ambient theory, itself classifying the geometric extension. This allows one to transfer factorial and topological aspects to categories of models in arbitrary (Grothendieck) toposes. Therefore, the spectral construction still makes sense at the level the category of modelled toposes, which gather all possible models regardless of their support. There, the spectrum functor turns out to be the left adjoint of a forgetful functor, unveiling its nature of free construction : it associates to models of the initial theory a "free sheaf of models" of the geometric extension.
In this presentation we will synthetize the main works currently existing on this subject. After discussing the factorial aspects and their topological interpretation, which are analyzed in the work of Anel [1], we will investigate the logical and syntactic significance of the spectral construction, following the interpretation given in Coste [3]. We will also make explicit the links between the mainstream method and the somewhat divergent yet handy construction of spectra by Diers [4] which relies on the notion of multiadjunction. We shall finally mention how the factorization structure manifests itself at the level of classifying toposes as explained in Dubuc [5] and Cole [2], while the synthetic work of Lurie [7] and the fundamental ideas of Hakim [6] will constitute our main thread.
References
[1] Anel, Mathieu. Grothendieck topologies from unique factorisation systems. arXiv:0902.1130v2. 22 Oct 2009
[2] Cole, J.C. The bicategory of Topoi and Spectra. http://emis.ams.org/journals/TAC/reprints/articles/25/tr25.pdf
[3] Coste, Michel. Localisation, spectra and sheaf representation. springer.com/content/pdf/10.1007/BFb0061820.pdf
[4] Diers, Yves. Une construction universelle des spectres et faisceaux structuraux. Communication in Algebra, 12(17),21412183 (1984).
[5] Dubuc, Eduardo. Axiomatic etal maps and a theory of spectrum. Journal of pure and Applied Algebra 149 (2000) 1545.
[6] Hakim, Monique. Topos Annelés et schémas relatifs.
[7] Lurie, Jacob. Derived Algebraic Geometry V: Structured Spaces
[8] Taylor, Paul. The trace factorisation of stable functors.

25 octobre 2018 à 10h30, Salle de réunion Fizeau
Brett McLean [Slides]
Partial functions, being specialised binary relations, have long been studied by algebraic logicians, using their usual methodology, terminology, and techniques. However, whilst in algebraic logic proper, relations provide semantics for logical formulas, the current popular justification for studying partial functions is to provide semantics for deterministic computer programs.
By considering collections of partial functions closed under some operations of our choosing—composition and intersection are examples—we obtain classes of algebraic structures. I will start by enumerating some of the most important operations, and the questions we ask about the classes of algebras. Then I will briefly summarise what is known, making the standard comparison with the research on algebras of binary relations.
The work from my PhD divides into four parts. The first is about complete representations—isomorphisms that turn any existing infima/suprema into intersections/unions. The second is about the finite representation property, true of a class if every finite algebra in the class is isomorphic to one on a finite base set. The third is about 'multiplace' partial functions, which take multiple arguments. The last part concerns certain partial operations on partial functions. These operations appear in the 'stack and heap' semantics for separation logic, a logic used for reasoning about the memory usage of computer programs.
I will explain the main results from each of these four parts, and give a little detail on the most important and interesting points in the proofs.

18 octobre 2018 à 10h30, Salle de conférences
Brett McLean [Slides]
Partial functions, being specialised binary relations, have long been studied by algebraic logicians, using their usual methodology, terminology, and techniques. However, whilst in algebraic logic proper, relations provide semantics for logical formulas, the current popular justification for studying partial functions is to provide semantics for deterministic computer programs.
By considering collections of partial functions closed under some operations of our choosing—composition and intersection are examples—we obtain classes of algebraic structures. I will start by enumerating some of the most important operations, and the questions we ask about the classes of algebras. Then I will briefly summarise what is known, making the standard comparison with the research on algebras of binary relations.
The work from my PhD divides into four parts. The first is about complete representations—isomorphisms that turn any existing infima/suprema into intersections/unions. The second is about the finite representation property, true of a class if every finite algebra in the class is isomorphic to one on a finite base set. The third is about 'multiplace' partial functions, which take multiple arguments. The last part concerns certain partial operations on partial functions. These operations appear in the 'stack and heap' semantics for separation logic, a logic used for reasoning about the memory usage of computer programs.
I will explain the main results from each of these four parts, and give a little detail on the most important and interesting points in the proofs.

11 octobre 2018 à 10h30, Salle de réunion Fizeau
Axel Osmond
Some of the most prominent dualities between spaces and algebras, such as Stonelike dualities for ordered structures or Grothendieck duality between schemes and rings, may be seen as instantiations of a common construction, namely a spectral construction. It consists primarily of a contravariant adjunction given by:
 A spectrum functor assigning to any model of an algebraic theory a certain site equipped with a structural sheaf. The site carries a topology implicitly encoded in the algebraic structure of each model and its place inside the ambient category. Under some conditions it is just a space, while in others a sheaf is needed. The sheaf encodes geometric and local behavior, that is, all the nondualizable information that remains.
 A left adjoint reconstructing each algebraic structure from the geometric information attached to the associated structured space, or in some cases, directly from its topology.
In the most general setting, such a construction takes place in a category of models of an essentially algebraic theory, that is, a locally finitely presentable category. It is produced by the presence of an orthogonality/lifting structure, whose restriction over finitely presented objects mimics a topological behavior, with notions of points, local objects, etale opens and covers. Dually, since finitely presented objects mirror the syntactic category of the underlying theory, those topological data define some extension of this theory by axioms of geometric logic.
The topological and geometric data produced through this process can be gathered into the so called spectrum associated to setvalued models of the theory. However this construction does not only concern setvalued models. The induced topology defines a certain subtopos of the classifying topos of the ambient theory, itself classifying the geometric extension. This allows one to transfer factorial and topological aspects to categories of models in arbitrary (Grothendieck) toposes. Therefore, the spectral construction still makes sense at the level the category of modelled toposes, which gather all possible models regardless of their support. There, the spectrum functor turns out to be the left adjoint of a forgetful functor, unveiling its nature of free construction : it associates to models of the initial theory a "free sheaf of models" of the geometric extension.
In this presentation we will synthetize the main works currently existing on this subject. After discussing the factorial aspects and their topological interpretation, which are analyzed in the work of Anel [1], we will investigate the logical and syntactic significance of the spectral construction, following the interpretation given in Coste [3]. We will also make explicit the links between the mainstream method and the somewhat divergent yet handy construction of spectra by Diers [4] which relies on the notion of multiadjunction. We shall finally mention how the factorization structure manifests itself at the level of classifying toposes as explained in Dubuc [5] and Cole [2], while the synthetic work of Lurie [7] and the fundamental ideas of Hakim [6] will constitute our main thread.
References
[1] Anel, Mathieu. Grothendieck topologies from unique factorisation systems. arXiv:0902.1130v2. 22 Oct 2009
[2] Cole, J.C. The bicategory of Topoi and Spectra. http://emis.ams.org/journals/TAC/reprints/articles/25/tr25.pdf
[3] Coste, Michel. Localisation, spectra and sheaf representation. springer.com/content/pdf/10.1007/BFb0061820.pdf
[4] Diers, Yves. Une construction universelle des spectres et faisceaux structuraux. Communication in Algebra, 12(17),21412183 (1984).
[5] Dubuc, Eduardo. Axiomatic etal maps and a theory of spectrum. Journal of pure and Applied Algebra 149 (2000) 1545.
[6] Hakim, Monique. Topos Annelés et schémas relatifs.
[7] Lurie, Jacob. Derived Algebraic Geometry V: Structured Spaces
[8] Taylor, Paul. The trace factorisation of stable functors.

2 octobre 2018 à 10h30, Salle de conférences
Mai Gehrke