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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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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
@ZhenLin: Yeah, I agree here. It was just my last attempt to get a response before the bounty expires. –  NikolajK Mar 31 '14 at 16:17

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