Vol. 82, No. 2, 1979

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Semi-universal maps and universal initial completions

Horst Herrlich and George Edison Strecker

Vol. 82 (1979), No. 2, 407–428

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.

Mathematical Subject Classification 2000
Primary: 18A35
Secondary: 18A32
Received: 1 May 1978
Published: 1 June 1979
Horst Herrlich
George Edison Strecker