NNOs in different toposes "the same"?

[David Roberts <david.roberts@adelaide.edu.au>]

Hi all,

If have a geometric morphism f: E -> F, what's the/a sensible way to
say that the natural number objects of E and F are 'the same'? If f is
local, then f_* preserves colimits, and so both f^* and f_* respect
natural numbers objects up to iso. But this is a little too strong,
perhaps, since we only need f_* to respect finite limits to use the
characterisation of |N by Freyd to show preservation. What other
conditions could I impose, other than simply that f_* preserves the
NNO?

Secondly, what if E is the externalisation of an internal topos in F?
For instance, F = Set and E the externalisation of a small topos, not
necessarily an internal universe (in fact I don't want this to be the
case!). Then if I can say what it means for the NNO in E to be 'the
same as' that in F, I can say that the internal topos has the same NNO
as the ambient category.

Regards,

David


-- 
Dr David Roberts
Research Associate
School of Mathematical Sciences
University of Adelaide
SA 5005
AUSTRALIA


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