The Michael completion of a topos spread (Q1850095)

The topos-theoretic analogue of the notion of spread, as introduced in the realm of topological spaces by \textit{R. H. Fox}, was studied by the first two authors of this paper [\textit{M. Bunge} and \textit{J. Funk}, J. Pure Appl. Algebra 113, 1-38 (1996; Zbl 0861.18004), ibid. 130, 49-84 (1998; Zbl 0924.18001)]. \textit{E. Michael} [Nederl. Akad. Wet., Proc., Ser. A 66, 629-633 (1963; Zbl 0202.22101)] introduced a different approach to Fox's spreads, which in particular resulted in a notion of completion which was subtly different from Fox's. The present paper shows that Michael's definition may also be re-interpreted for morphisms of toposes, and compares the Fox and Michael completions in this context.