On the Axioms of Arboreal Categories

Abstract

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.

26 pages. This is an extended version of a paper due to appear in the proceedings of the 18th International Workshop on Coalgebraic Methods in Computer Science (CMCS 2026)