Dialogue Categories and Frobenius Monoids

• Published in: Computation, Logic, Games, and Quantum Foundations - the Many Facets of Samson Abramsky Vol. 7860; pp. 197 - 224
• Main Author: Melliès, Paul-André
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2013; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Frobenius Algebra; Monoid Structure; Monoidal Category; Symmetric Monoidal Category; Tensorial Logic
• ISBN: 3642381634; 9783642381638
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-38164-5_15

About ten years ago, Brian Day and Ross Street discovered a beautiful and unexpected connection between the notion of ∗-autonomous category in proof theory and the notion of Frobenius algebra in mathematical physics. The purpose of the present paper is to clarify the logical content of this connection by formulating a two-sided presentation of Frobenius algebras. The presentation is inspired by the idea that every logical dispute has two sides consisting of a Prover and of a Denier. This dialogical point of view leads us to a correspondence between dialogue categories and Frobenius pseudomonoids. The correspondence with dialogue categories refines Day and Street’s correspondence with ∗-autonomous categories in the same way as tensorial logic refines linear logic.