**Title:** Adjointness in Modal Semantics

**Abstract:** Coalgebraic modal logic is a set of approached to modal semantics that defines frames and models as coalgebras for a functor. This approach has captured the traditional notions of Kripke and Neighbourhood frames, as well as many other types of semantic structures unfamiliar in traditional philosophical logic. In this talk I will present the coalgebraic perspective on Kripke and Neighbourhood frames, and show how this perspective can lead us in helpful new directions. Once we adopt a coalgebraic perspective, we can see that the central invariance result about modal definability—that all formulas of the basic modal language are invariant under bounded morphisms—follows from a single, fundamental fact about functions: that direct images are left adjoint to inverse images. However, there is also a right adjoint to inverse image: the “codirect image.” Using the right adjoint to inverse image, instead of the left, leads to another, non-traditional, modal language and another invariance result. This talk will explain how this adjoint sequence arises from a coalgebraic perspective on Kripke frames, and discuss its implications for defining modalities in the Neighbourhood frames.