From: Jonas Frey <jonas743@gmail.com>
To: Andrej Bauer <andrej.bauer@andrej.com>
Cc: categories list <categories@mta.ca>
Subject: Re: An internal definition in a realizability topos
Date: Wed, 26 Sep 2012 19:58:43 +0200 [thread overview]
Message-ID: <E1TGwc6-000054-1r@mlist.mta.ca> (raw)
In-Reply-To: <E1TGDAf-0007KR-Gt@mlist.mta.ca>
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/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
prev parent reply other threads:[~2012-09-26 17:58 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2012-09-24 15:52 Andrej Bauer
2012-09-24 19:02 ` Andrej Bauer
2012-09-26 17:58 ` Jonas Frey [this message]
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