  Arboreal categories, introduced by Abramsky and Reggio, axiomatise categories
with tree-shaped objects. These categories provide a categorical language for
formalising behavioural notions such as simulation, bisimulation, and
resource-indexing. In this paper, we strengthen the axioms of an arboreal
category to exclude `branching' behaviour, obtaining a notion of `linear
arboreal category'. We then demonstrate that every arboreal category satisfying
a linearisability condition has an associated linear arboreal subcategory
related via an adjunction. This identifies the relationship between the
pebble-relation comonad, of Montacute and Shah, and the pebbling comonad, of
Abramsky, Dawar, and Wang, and generalises it further. As another outcome of
this new framework, we obtain a linear variant of the arboreal category for
modal logic. By doing so we recover different linear-time equivalences between
transition systems as instances of their categorical definitions. We conclude
with new preservation and characterisation theorems relating trace inclusion
and trace equivalence with different linear fragments of modal logic.
