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Since the question did not get an answer here I have posted it to mathoverflow at

All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.

I would like to know whether the inverse image part of a geometric morphism always preserves W-types for more general reasons or pointers to references detailing the conditions on functors so they preserve W-types.

Inverse image functors always preserve the natural numbers object, which is a particular kind of W-type, but the proof of this fact is very specific to the NNO so this might suggest that they don't all preserve W-types, but I have not found anything about this stated anywhere.

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closed as off-topic by Aleš Bizjak, Solid Snake, PhoemueX, Zhen Lin, 91500 Sep 22 '15 at 9:33

  • This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

Initial algebras for the simplest polynomial endofunctors (i.e. those of the form $X \mapsto A_0 + A_1 \times X + A_2 \times X^2 + \cdots + A_n \times X^n$) can be constructed using only colimits and finite products, so they are preserved by any inverse image functor. But I imagine there are difficulties for the general case, because inverse image functors don't usually preserve exponential objects – so it's not even clear whether algebras get mapped to algebras. – Zhen Lin Nov 6 '13 at 21:55
@NiftyKitty95 You should not change "toposes" to "topoi". Let the writer decide. – Zhen Lin Mar 31 '14 at 14:00
I want to know the answer to this too. Since it didn't get any answers here, how about asking on MathOverflow? – Mike Shulman Sep 16 '15 at 2:44
@MikeShulman I have asked the question at mathoverflow. The link is…. What do I do with the question here? – Aleš Bizjak Sep 21 '15 at 16:23
I'm voting to close this question because I did not get the answer and I have asked the same question at mathoverflow now. – Aleš Bizjak Sep 21 '15 at 16:25