The Michael completion of a topos spread

Abstract

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 ϕ : F→ E between toposes bounded over a base topos S . The first is the nature of the domain of ϕ, which need only be assumed to be a “definable dominance” over S , a condition that is trivially satisfied if S is a Boolean topos. The Heyting algebras arising from the object Ω S of truth values in the base topos play a special role in that they classify the definable monomorphisms in those toposes. The geometric morphisms F→ F′ over E which preserve these Heyting algebras (and that are not typically complete) are said to be strongly pure. The second is the nature of ϕ 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 E correspond to the S -valued Lawvere distributions on E (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 “ S -additive measures” on E whose analysis we do not pursue here. We close the paper with several other open questions and directions for future work.