Abstract
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.
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References
Abramsky, S., Honda, K., McCusker, G.: A fully abstract game semantics for general references. In: Proceedings of the Thirteenth Annual IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press (1998)
Abramsky, S., Blute, R., Panangaden, P.: Nuclear and trace ideals in tensored ∗-categories. Journal of Pure and Applied Algebra 143, 3–47 (1999)
Abramsky, S., Heunen, C.: H ∗ -algebras and nonunital Frobenius algebras. Clifford Lectures, AMS Proceedings of Symposia in Applied Mathematics (2010) (to appear)
Barr, M.: ∗-autonomous categories. Lectures Notes in Mathematics, vol. 752. Springer (1979)
Curtis, C., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Pure and Applied Mathematics, vol. XI. Interscience Publishers, New York-London (1962)
Day, B., Street, R.: Quantum categories, star autonomy, and quantum groupoids. In: Galois Theory, Hopf Algebras, and Semiabelian Categories. Fields Institute Communications, vol. 43, pp. 193–231. American Math. Soc. (2004)
Egger, J.: The Frobenius relations meet linear distributivity. Theory and Applications of Categories 24(2), 25–38 (2010)
Girard, J.-Y.: Linear logic. Theoretical Computer Science, 50–102 (1987)
Honda, K., Tokoro, M.: An Object Calculus for Asynchronous Communication. In: America, P. (ed.) ECOOP 1991. LNCS, vol. 512, pp. 133–147. Springer, Heidelberg (1991)
Honda, K., Yoshida, N.: Game Theoretic Analysis of Call-by-Value Computation. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 225–236. Springer, Heidelberg (1997)
Joyal, A., Street, R.: An introduction to Tannaka duality and quantum groups. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds.) Proceedings of the Category Theory, Como 1990. Lecture Notes in Math., vol. 1488. Springer, Heidelberg (1991)
Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)
Kassel, C.: Quantum Groups. Graduate Texts in Mathematics, vol. 155. Springer (1995)
Melliès, P.-A.: Game semantics in string diagrams. In: Proceedings of the Annual ACM/IEEE Symposium on Logic in Computer Science (2012)
Melliès, P.-A.: Dialogue categories and chiralities (submitted, manuscript available on the author’s web page)
Melliès, P.-A.: A micrological study of helix negation (submitted, manuscript available on the author’s web page)
Melliès, P.-A.: Braided notions of dialogue categories (submitted, manuscript available on the author’s web page)
Shum, M.C.: Tortile tensor categories. Journal of Pure and Applied Algebra 93, 57–110 (1994)
Street, R.: Frobenius monads and pseudomonoids. J. Math. Phys. 45, 3930 (2004)
Street, R.: Quantum Groups: A Path to Current Algebra. Australian Mathematical Society Lecture Series. Cambridge University Press (2007)
Wehr, M.: Higher Dimensional Syntax. In: The Proceedings of the Category Theory in Computer Science (CTCS) Conference (1999)
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Melliès, PA. (2013). Dialogue Categories and Frobenius Monoids. In: Coecke, B., Ong, L., Panangaden, P. (eds) Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. Lecture Notes in Computer Science, vol 7860. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38164-5_15
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DOI: https://doi.org/10.1007/978-3-642-38164-5_15
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