TITLE: Is there a monad on Set whose algebras are Tychonoff spaces?
QUESTION [13 upvotes]: Compact Hausdorff spaces are algebras of the ultrafilter monad on Set.
Is the category of Tychonoff spaces also monadic over Set?

REPLY [4 votes]: Even easier.  Let $I$ be the unit interval and $E$ be the equivalence relation that identifies all the rationals.  $I/E$ is certainly not Hausdorff.  Is it a regular category?  I'm not sure.