Arboreal Categories: An Axiomatic Theory of Resources

Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop framework whereby the essential properties these categories are captured in purely axiomatic fashion. To end, introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation back-and-forth games, resource number rounds game, be defined. These related extensional or "static" structures via covers, resource-indexed comonadic adjunctions. ideas developed general, setting, applied relational structures, where constructions for pebbling, Ehrenfeucht-Fra\"iss\'e modal games recently introduced by Abramsky et al. recovered, showing that many fundamental theory descriptive complexity arise from instances covers.