Abstract

Presheaf categories have been used to make models of univalent type theory, first in classical set theory and later (with the “CCHM” model of Coquand’s group) in some form of constructive set theory. Here I concentrate on intuitionistic ZF set theory with atoms (IZFA). Back in 1980 it was noticed by Mike Fourman, Dana Scott and others that if one works in IZFA , then it is possible to regard presheaves (over a given index category) just as sets. At the same time, dependent type theory has a straightforward interpretation in set theory. Under this interpretation, what is needed for a universe of sets in IZFA to model univalence? Since univalence is a property of identity, to answer this question one first has to look at the set-theoretic version of (propositional) identity types. Restricting to identity types generated by an interval, I will discuss some simple axioms that suffice for universes of IZFA sets to be univalent and which incorporate set-theoretic versions of some observations of Lumsdaine, Shulman and others.