Abstract
We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. μ-bicomplete categories.We argue that parity games with a given starting position play the role of terms for the theory of μ-bicomplete categories. We show that the interpretation of a parity game in the category of sets and functions is the set of deterministic winning strategies for one player in the game.We discuss bounded memory communication strategies between two parity games and their computational significance. We describe how an attempt to formalize them within the algebra of μ-bicomplete categories leads to develop a calculus of proofs that are allowed to contain cycles.
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