Peter LeFanu Lumsdaine: Fraenkel–Mostowski models are continuous G-sets, or something like that

Abstract

Fraenkel–Mostowski permutation models are a straightforward but flexible technique for independence results in set theory, giving models of ZFA (set theory with atoms). Categorically, these can be viewed as arising from a topos of continuous G-sets, or alternatively as a slight modification of this (as noted by Fourman, Blass, and Scedrov).

Why are there two possible toposes corresponding to this model? What does the difference between them mean, either as toposes in their own right or in terms of their connection to set-theoretic Fraenkel–Mostowski models?

I’ll recall all of these constructions and compare them, largely following Blass–Scedrov, but partly also in the more recent light of Shulman’s stack semantics.