This will be a talk for the Logic Advanced Class at the Andrew Wiles Building in Oxford on the June 5th 2023 at 11 am local time. This presentation will draw upon a circulating preprint of my forthcoming article bearing the same title. The project has been supervised by Joel David Hamkins.

I shall explore Kripke categories—extensions of concrete categories that incorporate natural modal semantics, offering a fresh lens for the analysis of categories through modal logic. Central to my analysis will be a comprehensive examination of the archetypical category of sets, taking into account various naturally occurring classes of morphisms and elucidating their intrinsic propositional modal validities. Through this exploration, I hope to highlight the intriguing connections between category theory, model theory, and modal logic, opening doors for new thoughts and discussions in these fields.