# Logical endofunctors of Set?

What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. preserves finite limits and power objects—hence also finite colimits and exponentials)?

On the lower end, Andreas Blass proved ("Exact functors and measurable cardinals") that there exists a nontrivial exact endofunctor of Set (that is, preserving finite limits and colimits) iff there exists a measurable cardinal. Since logical functors are a fortiori exact, the existence of a measurable cardinal is a necessary condition. On the upper end, any nontrivial elementary embedding j:V→V surely induces a logical endofunctor of Set, so the existence of a Reinhardt cardinal is a sufficient condition. But can it be pinned down more precisely?

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Here is a comment I learned from Nate Ackermann. Blass shows that any exact endofunctor of Set is a directed union of ultrapowers by countably-complete ultrafilters. Suppose our functor were just a single countably-complete ultrapower. Then it would also give an elementary embedding j: V → M for some transitive model M ⊆ V containing all the ordinals. But if j also preserves powersets, then M must also contain all the subsets of the ordinals, which implies it is all of V. Hence no single ultrapower can be a logical functor. –  Mike Shulman Nov 16 '09 at 4:16
Strictly speaking, Joel is correct that I proved the result he cites, but it was proved earlier by Vera Trnkova (Comm. Math. Univ. Carolinae 12 (1971) 227-233). My paper on this (Pac. J. Math. 63 (1976) 335-346) was followed by a correction about this point (Pac. J. Math 73 (1977) 540). –  Andreas Blass Sep 29 '10 at 13:26