Categorical Combinatorics for Innocent Strategies

Russ Harmer (1) , Martin Hyland (2) , Paul-André Melliès (1)

1 PPS - Preuves, Programmes et Systèmes
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics

We show how to construct the category of games and innocent strategies from a more primitive category of games. On that category we define a comonad and monad with the former distributing over the latter. Innocent strategies are the maps in the induced two-sided Kleisli category. Thus the problematic composition of innocent strategies reflects the use of the distributive law. The composition of simple strategies, and the combinatorics of pointers used to give the comonad and monad are themselves described in categorical terms. The notions of view and of legal play arise naturally in the explanation of the distributivity. The category-theoretic perspective provides a clear discipline for the necessary combinatorics.

Domaines: Informatique et théorie des jeux [cs.GT]; Catégories et ensembles [math.CT]; Combinatoire [math.CO]

Russ Harmer, Martin Hyland, Paul-André Melliès. Categorical Combinatorics for Innocent Strategies. Logic in Computer Science, Jul 2007, France. pp. 379-388. ⟨hal-00150373⟩