  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 very
general, axiomatic setting, and applied to relational structures, where the
recently introduced comonadic constructions for pebbling,
Ehrenfeucht-Fra\"iss\'e and modal bisimulation games are recovered, showing
that many of the fundamental notions of finite model theory and descriptive
complexity arise from instances of arboreal covers.
