Algebra, Categories and Models - Reading Group

There is also some logic involved, if one really insists. Organized by Rodrigo Almeida and Lingyuan Ye .

Description

This reading group is centered on reading the book “Sheaves, Games and Model Completions”. The goal is to understand some of the central techniques exploited by the authors, especially Ghilardi, in proving and reproving some key theorems of non-classical logic: Pitts’ uniform interpolation theorem; the non-uniform interpolation of S4; Ruitenburg’s theorem; the Finitarity of Unification of IPC, S4, and other systems; the structure of algebras of truth-values in Elementary Toposes; amongst others. Since many of these topics are active areas of research, the focus of the reading group will be on topics immediately relevant to the participants’ interests and research specializations. Every two weeks we will meet with a presentation, which will give an overview of the reading material for those two weeks. Reading material will be updated regularly here at the end of each session.

Our first focus will be on getting the basic concepts of the field, before moving on to the key concepts:

  1. Free, Finitely Generated and Finitely Presented Algebras
  2. Congruence Properties, Amalgamation Properties
  3. Model Completions
  4. R-Regular Categories and R-Heyting Categories
  5. Heyting Algebras and the structure of Finitely Presented Heyting Algebras

References

Silvio Ghilardi, Marek Zawadowski. (2002). Sheaves, Games and Model Completions: A Categorical approach to Non-classical Propositional Logics.

Meeting Schedule

We meet bi-weekly. Meetings are in person in SP107.

Date Speaker Topic Reading Material Location Time
23/10/2023 Lide Grotenhuis Kick-Off Meeting: Introductions. Basic Motivations. Uniform Interpolation and Model Completions. Definition of Model Completion and Examples. Introduction. Preliminary Notions (pp.15-30). F2.11 11h00-13h00
06/11/2023 Daniel Otten Regular and R-Regular Categories: Motivations and Examples. Section 3.1 (Model Completions - R-Heyting Categories) (pp.49-60). F2.11 11h00-13h00
21/11/2023 Lingyuan Ye R-Heyting Categories, Model Completions from R-Heyting Categories. Section 3.2 (Model Completions and Finitely Presented Algebras) (pp.56-64). F2.11 11h00-13h00
12/12/2023 Johannes Kloibhoffer L-bisimulation Games and Introduction to the Duality Theory for Heyting Algebras. Section 4.1 and Beginning of 4.2 (Basic Definitions and Finitely Presented Heyting Algebras) (pp.73-78). F2.11 15h00-17h00
09/01/2024 Rodrigo Almeida Representations of Finitely Presented Heyting Algebras. Section 4.2 and Beginning of 4.3 (Finitely Presented Heyting Algebras and Duality for Heyting Algebras) (pp.78-84). F2.11 15h00-17h00
15/01/2024 Rodrigo Almeida Representations of Finitely Presented Heyting Algebras (continued). Section 4.2 and Beginning of 4.3 (Finitely Presented Heyting Algebras and Duality for Heyting Algebras) (pp.78-84). F2.11 15h00-17h00
29/01/2024 Lingyuan Ye Sheaf-theoretic dualities for Finitely Presented Heyting Algebras. Section 4.2 and Beginning of 4.3 (Finitely Presented Heyting Algebras and Duality for Heyting Algebras) (pp.78-84). See also Lingyuan's . excellent notes on this subject. F2.11 15h00-17h00
10/02/2024 Lide Grotenhuis Recap of Algebraic and Categorical Perspectives, and Logical Perspectives. Section 4.2 and Beginning of 4.3 (Finitely Presented Heyting Algebras and Duality for Heyting Algebras) (pp.78-84). See also Lingyuan's . excellent notes on this subject. Online 15h00-17h00
10/03/2024 Daniel Otten Model Completion of Heyting Algebras. Section 4.4 and 4.5 (except for the combinatorial lemma). Online 15h00-17h00
02/04/2024 Rodrigo Almeida The Combinatorial Lemma. Lemma 4.17 and Lemma 4.18. F2.11 15h00-17h00
16/04/2024 Lingyuan Ye Generalizing Sheaf-theoretic dualities to stack dualities. See Lingyuan's preprint. F2.11 15h00-17h00