Quantifiers and modalities

Analogy between logical properties of quantifiers and modalities was noticed long ago, and in contemporary logic there exist various translations of quantifiers into modalities and back. This talk focuses on correlation between $1$-variable modal and superintuitionistic predicate logics and propositional modal and intuitionistic modal logics. To study it, we apply simplicial semantics for predicate logics introduced by D. Skvortsov and methods from the theory of simplicial sets.