Categorical combinatorics for non deterministic strategies on simple games

Abstract

The purpose of this paper is to define in a clean and conceptual way a non-deterministic and sheaf-theoretic variant of the category of simple games and deterministic strategies. One thus starts by associating to every simple game a presheaf category of non-deterministic strategies. The bicategory of simple games and non-deterministic strategies is then obtained by a construction inspired by the recent work by Melliès and Zeilberger on type refinement systems. We show that the resulting bicategory is symmetric monoidal closed and cartesian. We also define a 2-comonad which adapts the Curien-Lamarche exponential modality of linear logic to the 2-dimensional and non deterministic framework. We conclude by discussing in what sense the bicategory of simple games defines a model of non deterministic intuitionistic linear logic.