An internal definition in a realizability topos

An internal definition in a realizability topos

[Andrej Bauer]

Consider a realizability topos RT(A) over some PCA A. There is an
embedding Nabla : Set -> RT(A), which takes a set X to the assembly
whose underlying set is X and the existence predicate is trivial,
i.e., every element of X is realized by every realizer of A.

Let [n] = {0, 1, 2, .., n-1} be the set with n elements. I have some
interest in the subobject of Nabla([n]) whose underlying set is [n]
and the existence predicate is

   E(0) = { numeral(0) }
   E(1) = { numeral(0), numeral(1) }
   ...
   E(k) = { numeral(k-1), numeral(k) }
   ...
   E(n-1) = {numeral(n-2)}

In words: there are n elements 0, 1, 2, ..., n-1, where two
consequtive elements share a realizer.

The question is; is this object definable in the internal language of the topos?

Note that Nabla(2) is definable as the object of not-not-stable truth
values, and that each Nabla(X) is also definable with a bit more work.

With kind regards,

Andrej


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Re: An internal definition in a realizability topos

[Andrej Bauer]

> Note that Nabla(2) is definable as the object of not-not-stable truth
> values, and that each Nabla(X) is also definable with a bit more work.

This should have read "each Nabla([n]) is definable".


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Re: An internal definition in a realizability topos

[Jonas Frey]

Dear Andrej,

the object that you are interested in has been studied by Jaap van
Oosten in his preprint

"A Notion of Homotopy for the Effective Topos"
www.staff.science.uu.nl/~ooste110/realizability/homtpyEff.pdf

A way to define it abstractly -- not using concepts that are
particular to realizability -- is as a
"gluing of intervals Nabla(2)", more precisely as the colimit of the diagram

Nabla(2) <- 1 -> Nabla(2) <- 1 -> Nabla(2) <- ... <- 1 -> Nabla(2).

I don't know if this counts as a "definition in the internal language"
for you, but certainly
the universal property can be expressed in the internal language, i.e.
there is a judgment
about cocones which holds iff the cocone is a colimit cocone.

Kind regards,

Jonas


On Mon, Sep 24, 2012 at 5:52 PM, Andrej Bauer <andrej.bauer@andrej.com> wrote:
> Consider a realizability topos RT(A) over some PCA A. There is an
> embedding Nabla : Set -> RT(A), which takes a set X to the assembly
> whose underlying set is X and the existence predicate is trivial,
> i.e., every element of X is realized by every realizer of A.
>
> Let [n] = {0, 1, 2, .., n-1} be the set with n elements. I have some
> interest in the subobject of Nabla([n]) whose underlying set is [n]
> and the existence predicate is
>
>    E(0) = { numeral(0) }
>    E(1) = { numeral(0), numeral(1) }
>    ...
>    E(k) = { numeral(k-1), numeral(k) }
>    ...
>    E(n-1) = {numeral(n-2)}
>
> In words: there are n elements 0, 1, 2, ..., n-1, where two
> consequtive elements share a realizer.
>
> The question is; is this object definable in the internal language of the topos?
>
> Note that Nabla(2) is definable as the object of not-not-stable truth
> values, and that each Nabla(X) is also definable with a bit more work.
>
> With kind regards,
>
> Andrej
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


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