Modal Logic of Transition Systems in the Topos of Trees

We investigate the adjunction between the category of tran-sition systems (with not nec. bounded morphisms) and the topos of treesS from the perspective of the basic modal logic K on transition systems.More specifically, we show how the modal logic K over an object of Sseen as a transition system can be lifted back as a subobject of S . Thisrelies on the usual mutually transpose formulations of modal satisfactionas either a map of transition system or as a map of Boolean algebras withoperators. Moreover, thanks to the usual geometric morphism from theBoolean toposPsh(|N∗|) of presheaves over the discrete category of nat-ural numbers to S , we can give a characterization of modal satisfactionwithin S as a map from formulae to total subobjects of S whose imagein the Boolean toposPsh(|N∗|) is an internal map of Boolean algebras.