On the Axioms of Arboreal Categories
This work addresses a foundational issue in mathematical logic and category theory, but it is incremental as it refines an existing axiomatic framework.
The authors tackled the inadequacy of the connectedness axiom in arboreal categories, a framework for game comonads in logic, by proposing tree-connectedness and showing that key properties remain valid, with the additional result that the path functor is a Street fibration.
Arboreal categories were introduced as an axiomatic framework for game comonads, which provide a comonadic view on many model-comparison games in logic. We demonstrate the inadequacy of the axiom stating that paths are connected. We then propose the notion of ``tree-connectedness'' to address this deficiency, and show that all the essential properties of arboreal categories that we are aware of remain valid under this new definition. Furthermore, we show that the path functor is a Street fibration.