An axiomatic approach to the multiverse of sets

Abstract

Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a simultaneous consideration of all set theoretical universes and the relationships between them, eliminating the need for recourse 'outside the theory' when carrying out constructions like forcing etc. We also explore multiversal category theory, showing that we are finally free of questions about 'largeness' at each stage of the categorification process when working in this theory -- the category of categories we consider for a given universe contains all large categories in that universe without taking recourse to a larger universe. We leverage this newfound freedom to define a category ${\bf Force}$ whose objects are universes and whose arrows are forcing extensions, a $2$-category $\mathcal{V}\mathfrak{erse}$ whose objects are the categories of sets in each universe and whose component categories are given by functor categories between these categories of sets, and a tricategory $\mathbb{Cat}$ whose objects are the $2$-categories of categories in each universe and whose component bicategories are given by pseudofunctor, pseudonatural transformation and modification bicategories between these $2$-categories of categories in each universe.