NNOs in different toposes "the same"?
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/ ]
On 2014-11-09, 10:42 PM, 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
Dear David, The short answer is: the inverse image functor f*:F -> E preserves NNO. The argument goes as follows: an NNO is an initial (1+)-algebra, that is, initial for the endofunctor [X] -> [1+X]. Since f* preserves 1 and +, it induces an isomorphism f*(1+) = (1+_)f*, which in turn induces a functor (f*)-alg: (1 +)- alg -> (1+)-alg Fact (++): The right adjoint f_* induces a right adjoint to f*-alg Hence (f*)-alg preserves initial objects, aka, NNO. The Fact (++) can be easily calculated, but a more formal argument is given as Thm A.5 in Hermida, Claudio, and Bart Jacobs. "Structural induction and coinduction in a fibrational setting." Information and Computation 145.2 (1998): 107-152. Regards, Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi Claudio yes, I know this. One can prove it from Freyd's characterisation of N by finite colimits, as I hinted in my first message. What I'm after is the other situation, when the direct image functor preserves the NNO. I believe the geometric morphism being local is enough, but much more than needed, so I was after weaker/alternative conditions. David On 11 November 2014 23:31, Claudio Hermida <claudio.hermida@gmail.com> wrote:
On 2014-11-09, 10:42 PM, 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
Dear David,
The short answer is: the inverse image functor f*:F -> E preserves NNO.
The argument goes as follows: an NNO is an initial (1+)-algebra, that is, initial for the endofunctor [X] -> [1+X]. Since f* preserves 1 and +, it induces an isomorphism f*(1+) = (1+_)f*, which in turn induces a functor
(f*)-alg: (1 +)- alg -> (1+)-alg
Fact (++): The right adjoint f_* induces a right adjoint to f*-alg
Hence (f*)-alg preserves initial objects, aka, NNO.
The Fact (++) can be easily calculated, but a more formal argument is given as Thm A.5 in
Hermida, Claudio, and Bart Jacobs. "Structural induction and coinduction in a fibrational setting." Information and Computation 145.2 (1998): 107-152.
Regards,
Claudio
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
Claudio Hermida -
David Roberts -
Peter Johnstone