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

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:

**Free, Finitely Generated and Finitely Presented Algebras****Congruence Properties, Amalgamation Properties****Model Completions****R-Regular Categories and R-Heyting Categories****Heyting Algebras and the structure of Finitely Presented Heyting Algebras**

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

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 |