Relating structure and power: Comonadic semantics for computational resources

Abstract Combinatorial games are widely used in finite model theory, constraint satisfaction, modal logic and concurrency theory to characterize logical equivalences between structures. In particular, Ehrenfeucht–Fraïssé games, pebble bisimulation play a central role. We show how each of these types can be described terms an indexed family comonads on the category relational structures homomorphisms. The index $k$ is resource parameter that bounds degree access underlying structure. coKleisli categories for give syntax-free characterizations wide range important equivalences. Moreover, coalgebras key combinatorial parameters: tree depth comonad, width pebbling comonad synchronization unfolding comonad. These results pave way systematic connections two major branches field computer science, which hitherto have been almost disjoint: categorical semantics algorithmic theory.