Categories of Coalgebraic Games with Selective Sum

Joyal categorical construction on (well-founded) Conway games and winning strategies provides a compact closed category, where tensor and linear implication are defined via Conway disjunctive sum (in combination with negation for linear implication). The equivalence induced on games by the morphisms coincides with the contextual closure of the equideterminacy relation w.r.t. the disjunctive sum. Recently, the above categorical construction has been generalized to non-wellfounded games. Here we investigate Joyalâ€TMs construction for a different notion of sum, i.e. selective sum. While disjunctive sum reflects the interleaving semantics, selective sum accommodates a form of parallelism, by allowing the current player to move in different parts of the board simultaneously. We show that Joyalâ€TMs categorical construction can be successfully extended to selective sum, when we consider alternating games, i.e. games where each position is marked as Left player (L) or Right player (R), that is only L or R can move from that position, R starts, and L/R positions strictly alternate. Alternating games typically arise in the context of Game Semantics. This category of well-founded games with selective sum is symmetric monoidal closed, and it induces exactly the equideterminacy relation. Generalizations to non-wellfounded games give linear categories, i.e. models of Linear Logic. Our game models, providing a certain level of parallelism, may be situated halfway between traditional sequential alternating game models and the concurrent game models by Abramsky and Mellies. We work in a context of coalgebraic games, whereby games are viewed as elements of a final coalgebra, and game operations are defined as final morphisms.