In this paper we investigate important categories lying strictly between the
Kleisli category and the Eilenberg-Moore category, for a Kock-Zöberlein monad
on an order-enriched category. Firstly, we give a characterisation of free
algebras in the spirit of domain theory. Secondly, we study the existence of
weighted (co)limits, both on the abstract level and for specific categories of
domain theory like the category of algebraic lattices. Finally, we apply these
results to give a description of the idempotent split completion of the Kleisli
category of the filter monad on the category of topological spaces.