Axioms and (counter)examples in synthetic domain theory (Q1577485)

The goal of the subject which has become known as synthetic domain theory is to provide an axiomatics for categories of domains which should allow one to reason about domains as if they were simply sets, their `non-set-like' features being masked by the internal logic of the category within which one is working. In the present paper, the authors present a careful development of synthetic domain theory based upon the internal logic of an elementary topos: they also interpret their definitions in several particular toposes (both realizability toposes and Grothendieck toposes), which enables them to give counterexamples to various conjectures about synthetic domain theory which have been made in the past, for example that the initial lift-algebra is necessarily an internal `countable colimit'.