Models of set-theory from elementary 2-topoi

There is a long tradition of comparing elementary topos theory with traditional $\in$-based set theories such as ZF, beginning with [1, 5, 6]. These two kinds of set theories are well-known to be “mismatched”, in that ZF allows unbounded quantification over sets, whereas this is not possible in the internal logic of a topos. The result is that, from an elementary topos, one can construct a model of only a weak fragment of ZF, in which only bounded quantification is allowed.

Two approaches that overcome this limitation are the “Algebraic Set Theory” of Joyal and Moerdijk [4] and the more recent “Stack Semantics” of Shulman [7]. I will explain a different approach, related to these two, to obtain models of set theory “topos-theoretically”, based on the notion of elementary 2-topos introduced in [8], and which I have been further developing in [2, 3] following ideas of M. Makkai and B. Boshuk.

The notion of 2-topos is an axiomatization of the basic elementary properties of the 2-category of categories, much as the notion of 1-topos is related to the category of sets. The central axiom stipulates that a 2-topos has a special object $𝒮$, playing the role of the 1-category of sets. Thus, 2-topos theory is akin to class-set theory, the objects of the 2-topos being the “classes” (or rather, “large categories”), and the objects of $𝒮$ being the “sets”. As I will explain, from the rather simple and natural axioms of an elementary 2-topos, one can produce a model of full (intuitionistic) ZF set theory. This suggests 2-topoi as a natural environment for the algebraic study of set-theoretic universes.