Parametrized Fixed Points and Their Applications to Session Types

Parametrized fixed points are of particular interest to denotational semantics and are often given by “dagger operations” [Stephen L. Bloom and Zoltán Ésik, Fixed-Point Operations on ccc's. Part I, Theoretical Computer Science (ISSN 0304-3975) 155 (1996), 1–38, https://doi.org/10.1016/0304-3975(95)00010-0; Stephen L. Bloom and Zoltán Ésik, Iteration Theories. The Equational Logic of Iterative Processes, in: EATCS Monographs on Theoretical Computer Science, Springer-Verlag Berlin Heidelberg, ISBN 978-3-642-78034-9, 1993, xv+630 pp., https://doi.org/10.1007/978-3-642-78034-9; Stephen L. Bloom and Zoltán Ésik, Some Equational Laws of Initiality in 2CCC's, International Journal of Foundations of Computer Science 6 (1995) 95–118, https://doi.org/10.1142/S0129054195000081.]. Dagger operations that satisfy the Conway identities [Stephen L. Bloom and Zoltán Ésik, Fixed-Point Operations on ccc's. Part I, Theoretical Computer Science (ISSN 0304-3975) 155 (1996), 1–38, doi: https://doi.org/10.1016/0304-3975(95)00010-0.] are particularly useful, because these identities imply a large class of identities used in semantic reasoning. We generalize existing techniques to define dagger operations on ω-categories and on O-categories. These operations enjoy a 2-categorical structure that implies the Conway identities. We illustrate these operators by considering applications to the semantics of session-typed languages.