Two Day Workshop for Category Theory and Logic

12 - 13th March, 2009, Kyoto, Japan

Lecture Notes

• Tohoku/Kyoto Workshop Lecture Notes are available now!

Photos

Location

• Research Building No. 2 1F Room 10, Yoshida Main Campus, Kyoto University, Kyoto, Japan

As for Research Building No. 2, check also Pictorial Map of Yoshida Campus

Program

12th (Thr) March [Morning Session]

• 10:00 -- 10:50: Tutorial on Category Theory and Categorical Logic 1
  Steve Awodey (CMU)
• 11:00 -- 11:50: Tutorial on Category Theory and Categorical Logic 2: Kripke semantics for lambda calculus
  Steve Awodey (CMU)
• 11:50 -- 13:00: Lunch Time

12th (Thr) March [Afternoon Session]

• 13:00 -- 13:40: Structuralism and Category Theory
  Minao Kukita (Kyoto University)
• 13:50 -- 14:30: An intuitionistic linear-time temporal next operator
  Kensuke Kojima (Kyoto University)
• 15:00 -- 15:40: Generalizations of Stone Dualities via the Notion of Topological Dualizability
  Yoshihiro Maruyama (Kyoto University)
• 15:50 -- 16:30: Proofs as logical knots on cobordism
  Paul-Andre Mellies (CNRS, Paris 7)
• 18:00 --: Reception

13th (Fri) March [Morning Session]

• 9:30 -- 10:20: Tutorial on Categorical Semantics of Various Logics 1: A comparison of different foundational systems
  Steve Awodey (CMU)
• 10:30 -- 11:20: Tutorial on Categorical Semantics of Various Logics 2: A comparison of different foundational systems
  Steve Awodey (CMU)

Abstracts

• Structuralism and Category Theory
  Minao Kukita (Kyoto University)
  

  Category theorists such as Mac Lane and Awodey argue that category theory provides us with better tools for studying mathematical structures than set theory or symbolic logic does. In doing so, however, they do not claim to give some philosophical foundation of mathematics.

  On the other hand, philosophical structuralists such as Putnam or Hellman intend to answer some basic philosophical (ontological or epistemological) questions about mathematics. Yet their explanations for mathematical objects and structures are not satisfactory particularly because of their discontinuity from mathematical practice.

  In this talk I describe what insight category theory brings to philosophy of mathematics, and then propose a philosophical view of mathematical objects and structures that I hope is more consonant with at least some part of mathematical practice such as category theory or theoretical computer science.

• An intuitionistic linear-time temporal next operator
  Kensuke Kojima (Kyoto University)
  

  We discuss Kripke semantics for linear-time temporal next operator in an intuitionistic setting. Motivated by a computational interpretation derived by Curry-Howard correspondence, we consider the converse of the axiom K (CK, for short) as an axiom characterizing the next operator. So the logic we study is actually an intuitionistic modal logic augmented by the axiom CK. To give a Kripke semantics for this logic, we consider Kripke frames with two accessibilities, one for modal and the other for intuitionistic counterpart, and investigate a frame condition corresponding to CK. It turns out that there is a gap between the intuitive meaning of "linear-time" and the actual frame condition. This result implies a subtle difference between classical and intuitionistic modal logics.

• Generalizations of Stone Dualities via the Notion of Topological Dualizability
  Yosihhiro Maruyama (Kyoto University)
  

  In this talk, we consider universal algebraic extensions of Stone dualities for distributive lattices and for Heyting algebras. In 1971, T. K. Hu generalized Stone duality for Boolean algebras to a universal duality for the quasi-variety generated by any primal algebra. By introducing the notion of topological dualizability, which extends that of primalness, we generalize Stone duality for distributive lattices to a universal duality for the quasi-variety generated by any finite ordered algebra which is topologically dualizable with respect to the Alexandrov topology. By this duality, we obtain dualities for some residuated quasi-varieties, including the class of Heyting algebras and the class of algebras of an intuitionistic many-valued logic defined by many-valued Kripke semantics.

• Proofs as logical knots on cobordism
  Paul-Andre Mellies (CNRS, Paris 7)
  

  My main ambition in this introductory talk will be to explain the topological nature of game semantics, and to clarify its relationship to a basic notion of cobordism studied in elementary particle physics.

Links

• Tohoku/Kyoto Workshop Lecture Notes by Steve Awodey.
• Sendai Logic and Philosophy seminar, March 2009
• Category Theory, Oxford Logic Guides, Oxford University Press, 2006 by Steve Awodey.

Contact Information

Katsuhiko Sano
Graduate School of Letters, Kyoto University
Yoshida Hommachi, Sakyo-ku, Kyoto, 606-8501, JAPAN
Email: