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