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μ-Bicomplete Categories and Parity Games

Published online by Cambridge University Press:  15 December 2002

Luigi Santocanale
Luigi Santocanale*
Affiliation:
LaBRI, Université Bordeaux 1, Domaine Universitaire, 351 cours de la Libération, 33405 Talence Cedex, France; [email protected].
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Abstract

For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by μ-terms. We call the category μ-bicomplete if every μ-term defines a functor. We provide concrete examples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved through parity games: we associate to each game an algebraic expression and turn the game into a term of a categorical theory. We show that μ-terms and parity games are equivalent, meaning that they define the same property of being μ-bicomplete. Finally, the interpretation of a parity game in the category of sets is shown to be the set of deterministic winning strategies for a chosen player.

Keywords

Parity gamesbicomplete categoriesinitial algebrasfinal coalgebrasinductive and coinductive types.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 36 , Issue 2: Fixed Points in Computer Science (FICS'01) , April 2002 , pp. 195 - 227
Copyright
© EDP Sciences, 2002

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