Ultrapowers and reduced powers are two popular tools for studying countable (and separable metric) structures. Once an ultrafilter on N is fixed, these constructions are functors into the category of countably saturated structures of the language of the original structure. The question of the exact
relation between these two functors has been raised only recently by Schafhauser and Tikuisis, in the
context of Elliott’s classification programme. Is there an ultrafilter on N such that the quotient map
from the reduced product associated with the Fréchet filter onto the ultrapower has the right inverse?
The answer to this question involves both model theory and set theory.
Although these results were motivated by the study of C*-algebras, all of the results and proofs will
be given in the context of classical (discrete) model theory.