From Strategies to Profunctors

A lax functor from a bicategory of very general nondeterministic concurrent strategies on concurrent games to the bicategory of profunctors is presented. The lax functor provides a fundamental connection between two approaches to generalizations of domain theory to forms of intensional domain theories, one based on game and strategies, and the other on presheaf categories and profunctors. The lax functor becomes a pseudo functor on the sub-bicategory of rigid strategies which includes ‘simple games’ (underlying AJM and HO games) and stable spans (specializing to Berry’s stable functions, in the deterministic case). In general, the lax functor illustrates how composition of strategies is obtained from that of profunctors by restricting to ‘reachable’ elements. The results are based on a new characterization of concurrent strategies, which exhibits concurrent strategies as certain discrete fibrations, or equivalently presheaves, over configurations of the game. Finally, the characterization suggests how to extend the definition of strategy to that of strategy on and between categories with a factorization system, an idea that relates to earlier work on bistructures and bidomains.