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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]