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
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