Linear Arboreal Categories
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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. Comonadic adjunctions between an arboreal category and any extensional category of objects, called arboreal covers, allow for application of the behavioural notions to the static objects of the extensional category. In this paper, we demonstrate that every arboreal category has an associated linear arboreal subcategory which admits a categorical version of trace equivalence and complete trace equivalence. This generalises the connection between the pebble-relation comonad, of Montacute and Shah, and the pebbling comonad, of Abramsky, Dawar, and Wang. Specialising to the pebble-relation comonad, we obtain categorical semantics for equivalence in a restricted conjunction fragment of infinitary finite variable logic. As another example, we construct a linear modal comonad recovering trace inclusion, trace equivalence, and complete trace equivalence between transition systems as instances of their categorical definitions. We conclude with a new Rossman preservation theorem relating trace inclusion with trace equivalence, and a van Benthem characterisation theorem relating complete trace equivalence with a linear fragment of modal logic.
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