Initial completions of categories
(A,U) over a base category are investigated simultaneously with certain
generalizations of the notion of topological functor. The main result states that
(A,U) has a reflective universal initial completion if and only if the functor U is
topologically algebraic in the sense of Y. H. Hong. This is analogous to results
of Hoffmann, Tholen, and Wischnewsky that (A,U) has a reflective Mac
Neille completion if and only if U is semi-topological. In addition, the class
of semi-topological functors is shown to be the smallest class closed under
composition and containing all topologically algebraic functors. It is also
shown that for any (E,M)-functor U (resp. (E,M)-category) E must be
contained in the class of generating U-morphisms (resp. epimorphisms). Specific
constructions of the above completions are given, the first necessitating the new
concept of semi-universal morphism. Examples illuminating the theory are also
provided.
