Johnstone's theorem and Fox completions

Marta Bunge

In ``Schetches of an Elephant: A compendium of topos theory'', Oxford University Press, 2002, Peter Johnstone proves a new general result (Theorem C3.3.14) describing the pullback topology along bounded geometric morphisms in a surprisingly simple manner, although its proof is both subtle and difficult.

One such pullback had been employed (Bunge and Funk, ``Spreads and the symmetric topos'' (SST), JPAA 113:1-38, 1996) in order to define in topos theory the notion of a complete spread, and to prove the equivalence between complete spreads over and distributions on a topos. However, the description of the topology (which strictly speaking was not needed in our categorical treatment) was sketchy. Johnstone's result has now become useful in my identification of an ingredient which renders the complete spread topology explicit. This ingredient is none other than a topos version of the Fox completion of a spread (R.H.Fox, ``Covering spaces with singularities'', Princeton, 1957, and E. Michael,``Cuts'', Acta Math. 111:1-36, 1964). The result that I have just obtained then brings around the motivating idea for the work in SST and completes a full circle of ideas. This will be the subject matter of my lecture.

I will also mention in this connection that a different (but clearly related) aspect of an application of Johnstone's theorem to the complete spreads defining pullback of SST is to obtain an elementary (in the language of categories fibred over a base topos) notion of (definable) completeness for geometric morphisms, on a par with (but opposite to) the existing elementary notion of a spread from SST. This second aspect was originally undertaken as joint project with J.Funk, M. Jibladze and T. Streicher.

File translated from TEX by TTH, version 3.01.
On 13 Feb 2004, 16:58.