Strategies as Profunctors

A new characterization of nondeterministic concurrent strategies exhibits strategies as certain discrete fibrations—or equivalently presheaves—over configurations of the game. This leads to a lax functor from the bicategory of strategies to the bicategory of profunctors. The lax functor expresses how composition of strategies is obtained from that of profunctors by restricting to ‘reachable’ elements, which gives an alternative formulation of the composition of strategies. It provides a fundamental connection—and helps explain the mismatches—between two generalizations of domain theory to forms of intensional domain theories, one based on games and strategies, and the other on presheaf categories and profunctors. In particular cases, 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), the lax functor becomes a pseudo functor. More generally, the laxness of the functor suggests what structure is missing from categories and profunctors in order that they can be made to support the operations of games and strategies. By equipping categories with the structure of a ‘rooted’ factorization system and ensuring all elements of profunctors are ‘reachable,’ we obtain a pseudo functor embedding the bicategory of strategies in the bicategory of reachable profunctors. This shift illuminates early work on bistructures and bidomains, where the Scott order and Berry’s stable order are part of a factorization system, giving a sense in which bidomains are games.