The paper brings together several lines of research, including the categorical perspective on model comparison games and the associated logical equivalences, the notion of arboreal categories, the linear variant of the pebbling comonad, and the linear-time branching-time spectrum of behavioural equivalences.
The key contributions are:
The definition of linear arboreal categories, which strengthen the axioms of arboreal categories to exclude 'branching' behaviour. This allows for the study of 'linear' variants of previously examined arboreal categories.
The construction of a linear arboreal subcategory CL from any linearisable arboreal category C, related via an adjunction I ⊣ T.
The definition of linear behavioural relations (e.g. trace inclusion, labelled trace equivalence) on objects in an extensional category E, using the linear arboreal cover Lk ◦ Ik ⊣ Tk ◦ Rk derived from an arboreal cover Lk ⊣ Rk of E.
New preservation and characterisation theorems relating linear-time behavioural equivalences (e.g. labelled trace equivalence) with linear fragments of logics captured by the branching equivalence (e.g. modal logic).
The paper provides a general framework for studying linear-time logics and equivalences, and demonstrates how this framework recovers and generalises several existing results in the literature.
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