As Claudio has said, inverse image functors always preserve the NNO, but David's question was: when does the direct image f_* preserve the NNO? A sufficient condition for this which is weaker than being local is that f should be connected, i.e. that f^* should be full and faithful, since then the unit map N -> f_*f^*N is an isomorphism. But this is certainly not necessary: for example, the inclusion from sheaves to presheaves on any locally connected internal site in a topos preserves N (see C3.3.10 in the Elephant). I don't know any necessary and sufficient condition (other than the condition that f_* preserves N!); if you restrict to countably cocomplete toposes (where N is always a countable copower of 1), then f_* preserves N iff it preserves all countable coproducts, iff f^* is full and faithful on morphisms with codomain N. But it's not clear to me what significance this latter condition has. Peter Johnstone On Mon, 10 Nov 2014, David Roberts wrote:
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
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