$μ$-Bicomplete Categories and Parity Games

Abstract: For an arbitrary category, we consider the least class of functors con- taining 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 $\mu$-terms. We call the category $\mu$-bicomplete if every $\mu$-term defines a functor. We provide concrete ex- amples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved through par- ity games: we associate to each game an algebraic expression and turn the game into a term of a categorical theory. We show that $\mu$-terms and parity games are equivalent, meaning that they define the same property of being $\mu$-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.