Concurrent Games over Relational Structures: The Origin of Game Comonads

TL;DR

This paper introduces a categorical framework for Spoiler-Duplicator games in finite model theory, using event structures and comonads to unify and generalize existing approaches, providing new insights into game strategies.

Contribution

It develops a systematic method to construct game comonads for all one-sided Spoiler-Duplicator games via adjunctions and bicategories, unifying strategies and homomorphisms.

Findings

Constructed game comonads from bicategories of game schemas.

Showed strategies in one-sided games coincide with generalized homomorphisms.

Characterized strategies in two-sided games as spans of event structures.

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.