Equivalences and Congruences on Infinite Conway's Games

Taking the view that infinite plays are draws, we study Conway non-terminating games and non-losing strategies. These admit a sharp coalgebraic presentation, where non-terminating games are seen as a final coalgebra and game contructors, such as disjunctive sum, as final morphisms. We have shown, in a previous paper, that Conway's theory of terminating games can be rephrased naturally in terms of game (pre)congruences. Namely, various conceptually independent notions of equivalence can be defined and shown to coincide on Conway's terminating games. These are the equivalence induced by the ordering on surreal numbers, the contextual equivalence determined by observing what player has a winning strategy, Joyal's categorical equivalence, and, for impartial games, the denotational equivalence induced by Grundy semantics. In this paper, we discuss generalizations of such equivalences to non-terminating games and non-losing strategies. The scenario is even more rich and intriguing in this case. In particular, we investigate efficient characterizations of the contextual equivalence, and we introduce a category of fair strategies and a category of fair pairs of strategies, both generalizing Joyal's category of Conway games and winning strategies. Interestingly, the category of fair pairs captures the equivalence defined by Berlekamp, Conway, Guy on loopy games.