  We prove that there exist profinite Heyting algebras that are not isomorphic
to the profinite completion of any Heyting algebra. This resolves an open
problem from 2009. More generally, we characterize those varieties of Heyting
algebras in which profinite algebras are isomorphic to profinite completions.
It turns out that there exists largest such. We give different
characterizations of this variety and show that it is finitely axiomatizable
and locally finite. From this it follows that it is decidable whether in a
finitely axiomatizable variety of Heyting algebras all profinite members are
profinite completions. In addition, we introduce and characterize representable
varieties of Heyting algebras, thus drawing connection to the classical problem
of representing posets as prime spectra.
