Duality Theory

Spring Meeting 2018

Duality Theory

In 1936, Marshall Stone discovered a representation for all Boolean algebras, obtained by using topological spaces, which gave algebraists a usable understanding of their structure. During the same years, two similar results were proved, by Garrett Birkhoff and by Lev Pontryagin, respectively for the class of finite distributive lattices and for the class of abelian groups. These three results planted the seeds for what is nowadays called Duality Theory. The beauty and the strength of this theory relies on the fact that it translates algebraic problems into dual topological problems, and this results in the possibility of making use of our geometric intuitions to study and tackle problems stated in abstract symbolic language.

Abstracts and Slides

Duality for distributive lattices with additional operations: a tutorial in two parts

Mai Gehrke and Sam van Gool

The mathematical theory of Stone duality underlies a deep connection between syntax and semantics in logic and theoretical computer science, and allows for powerful applications in both of these research fields. In particular, duality for distributive lattices provides a canonical notion of ‘model’ for any logical system with a distributing conjunction and disjunction. Duality theory for additional operations then helps to understand how the different connectives of the logic are reflected in these models. As an important classical example, Kripke semantics for intuitionistic and modal logics can be understood in this way. More recently, in joint work with V. Marra, we apply these ideas to multi-valued Lukaciewicz logic.
In this two-part tutorial, we will introduce duality for distributive lattices and additional operations from scratch, starting from the simple case of finite distributive lattices, and ending with full Priestley duality for expansions of distributive lattices. Along the way, we will illustrate the theory by showing applications to some of the logics mentioned above.

Sheaf representations and duality

Mai Gehrke

A sheaf of algebras is a structure half-way between algebra and topology. It allows the dualization of part of the structure of an algebra, while the remainder of the structure, residing in the `stalks’ of the sheaf, remains algebraic. This allows one to represent algebras as `continuously varying’ versions of simpler algebras: e.g. commutative rings with identity may be een as continuously varying `local rings’ and MV-algebras (the algebraic counterpart of Łukasiewicz infinitary propositional logic) may be seen as continuously varying MV-chains. It has long been known in universal algebra that any distributive sublattice of pairwise permuting congruences of an algebra yields a sheaf representation of the algebra. In joint work with Sam van Gool [1], we provide a generalization of this fact and prove a converse of the generalization. A central contribution of our work has been to identify stably compact spaces and the notion of softness for sheaves as central ingredients in getting a bijective correspondence. Stably compact spaces were first identified by Nachbin and are closely related to compact partially ordered spaces. They provide a non-Hausdorff counterpart to compact Hausdorff spaces. The notion of softness of sheaves originated with Godement’s treatment of homological algebra. In the special case where the algebras we want to represent have a distributive lattice reduct, we obtain a fully dualized description of sheaf representations over stably compact spaces as certain continuous bundles.
Our work grew out of our work on sheaf representations of MV-algebras with Marra [2] and as such it is closely related to recent work on sheaf representations for MV-algebras and ℓ-groups.

[1] M. Gehrke, S. J. v. Gool, Sheaves and duality. Journal of Pure and Applied Algebra 222 (2018) 2164–2180.

[2] M. Gehrke, S. J. v. Gool, V. Marra, Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality. Journal of Algebra 417 (2014) 290–332.


Type Refinement Systems and Duality

Paul-André Melliès

In this introductory and survey talk, I will review a number of elements of my ongoing work with Noam Zeilberger on type refinement systems and duality. I will start by explaining the meaning of the Isbell duality between the categories [C,Set] and [C^op,Set] of covariant and contravariant presheaves on a small category C. This duality will be then extended to the more general setting of type refinement systems [1]. The relevance of the construction will be illustrated by a number of fundamental and subtle issues in proof theory and substructural logics. If time permits, this trail will eventually lead us to the bifibrational reconstruction of Lawvere's presheaf hyperdoctrine in [2], following the philosophy of chiralities initiated and advocated in [3].

[1] Paul-André Melliès and Noam Zeilberger. An Isbell Duality Theorem for Type Refinement Systems. Math. Struct. in Comp. Science (2018), vol. 28, pp. 736-774.

[2] Paul-André Melliès and Noam Zeilberger. A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine. Proceedings of LICS 2016: 555-564

[3] Paul-André Melliès. Dialogue Categories and Chiralities Publications of the Research Institute for Mathematical Sciences, Volume 52, Issue 4, 2016, pp. 359–412.


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