  The closure of chains of embedding-projection pairs (ep-pairs) under bilimits
in some categories of predomains and domains is standard and well-known. For
instance, Scott's $D_\infty$ construction is well-known to produce directed
bilimits of ep-pairs in the category of directed-complete partial orders, and
de Jong and Escard\'o have formalized this result in the constructive domain
theory of a topos. The explicit construcition of bilimits for categories of
predomains and partial maps is considerably murkier as far as constructivity is
concerned; most expositions employ the constructive taboo that every
lift-algebra is free, reducing the problem to the construction of bilimits in a
category of pointed domains and strict maps. An explicit construction of the
bilimit is proposed in the dissertation of Claire Jones, but no proof is given
so it remained unclear if the category of dcpos and partial maps was closed
under directed bilimits of ep-pairs in a topos. We provide a
(Grothendieck)-topos-valid proof that the category of dcpos and partial maps
between them is closed under bilimits; then we describe some applications
toward models of axiomatic and synthetic domain theory.
