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stub for arithmetic pretopos, just to record the reference
We have the grey link arithmetic universe at abstract Stone duality, which we may want to redirect to arithmetic pretopos. I only don’t do it, since these are not the same thing (an AU has been defined by Joyal, though no one has seen it), even though Maietti’s proposal is that an arithmetic universe be defined as a list-arithmetic pretopos.
I hope we can flush Joyal’s definition out of the undergrowth…
I have added to arithmetic pretopos the redirect and the sentence
Maietti proposed that arithmetic pretoposes serve as the arithmetic universes that Andre Joyal once suggested to use for discussion of incompleteness theorems.
Alright :-). I guess there’s not much to discuss…
Trying to unwind Maietti’s definition of parameterised list objects, I guess they are just W-types for the non-locally cartesian closed version of the polynomial functor giving list objects. Cockett (JPAA 1990) defines them as initial algebras for the endofunctor $B+A\times (-)$ (for arbitrary $A$ and $B$).
Let me edit that in now… (EDIT: done!)
Thanks!
Gave hyperlinks to you finite products and (more importantly) disjoint coproducts.
Thanks very much Thomas! Fantastic that somebody has written up the details of this, I and many people have long wished to see the argument. I’d like to work through it when I get the chance and put some details on the nLab.
See here for an old nForum discussion around this.
Re #7: I’ve just posted something on this at MO.
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