# Johnstone's theorem and Fox completions

### Marta Bunge

Abstract

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.

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On 13 Feb 2004, 16:58.