The Michael completion of a topos spread

We continue the investigation of the extension into the topos realm of the concepts introduced by Fox (Cahiers Top. et Géometrie Diff. Catégoriques 36 (1995) 53) and Michael (Indag. Math. 25 (1963) 629) in connection with topological singular coverings. In particular, we construct an analog of the Michael completion of a spread and compare it with the analog of the Fox completion obtained earlier by the first two named authors (J. Appl. Algebra 113 (1996) 1). Two ingredients are present in our analysis of geometric morphisms image between toposes bounded over a base topos image. The first is the nature of the domain of phi, which need only be assumed to be a “definable dominance” over image, a condition that is trivially satisfied if image is a Boolean topos. The Heyting algebras arising from the object image of truth values in the base topos play a special role in that they classify the definable monomorphisms in those toposes. The geometric morphisms image over image which preserve these Heyting algebras (and that are not typically complete) are said to be strongly pure. The second is the nature of phi itself, which is assumed to be some kind of a spread. Applied to a spread, the (strongly pure, weakly entire) factorization obtained here gives what we call the “Michael completion” of the given spread. Whereas the Fox complete spreads over a topos image correspond to the image-valued Lawvere distributions on image (Acta Math. 111 (1964) 14) and relate to the distribution algebras (Adv. Math. 156 (2000) 133), the Michael complete spreads seem to correspond to some sort of “image-additive measures” on image whose analysis we do not pursue here. We close the paper with several other open questions and directions for future work.