Re: NNOs in different toposes "the same"?

[Peter Johnstone <ptj@dpmms.cam.ac.uk>]

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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