🤖 AI Summary
This paper addresses one-sided Spoiler-Duplicator games, aiming to establish a categorical semantics for strategies via game-theoretic models based on event structures.
Method: It uniformly models diverse one-sided games as event structures; constructs the game comonad—specifically, the game comonoid (i.e., comonoidal structure)—within the double category of signed games via universal construction; defines strategies as functors between game categories using adjunctions; and generalizes the notion of strategy to event-structure stacks, accommodating bidirectional scenarios.
Contributions: (1) It establishes a systematic construction paradigm for comonoids induced by one-sided games; (2) it proves that the resulting strategy functors are equivalent to generalized homomorphisms in the one-sided case and to event-structure stacks in canonical two-sided cases; (3) it provides a novel categorical semantic framework for finite model theory, grounded in event structures and double-category theory.
📝 Abstract
Spoiler-Duplicator games are used in finite model theory to examine the expressive power of logics. Their strategies have recently been reformulated as coKleisli maps of game comonads over relational structures, providing new results in finite model theory via categorical techniques. We present a novel framework for studying Spoiler-Duplicator games by viewing them as event structures. We introduce a first systematic method for constructing comonads for all one-sided Spoiler-Duplicator games: game comonads are now realised by adjunctions to a category of games, generically constructed from a comonad in a bicategory of game schema (called signature games). Maps of the constructed categories of games are strategies and generalise coKleisli maps of game comonads; in the case of one-sided games they are shown to coincide with suitably generalised homomorphisms. Finally, we provide characterisations of strategies on two-sided Spoiler-Duplicator games; in a common special case they coincide with spans of event structures.