Topos Semantics for Higher-Order Modal Logic

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E. In contrast to the well-known in- terpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier E, but rather by a suit- able complete Heyting algebra H. The canonical map relating H and E both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from surjec- tive geometric morphisms f : F → E, where H = f∗F. The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are not longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf seman- tics for propositional and first-order modal logic are subsumed by this notion.