Re: Cartesian bicategories and realizability

[flawler@scss.tcd.ie]

James Lipton wrote:
> This sounds a bit like the construction of realizability toposes in
> Categories and Allegories (Freyd and Scedrov)

Thanks.  I don't have access to that book at the moment, but I seem to
remember that they don't mention triposes.  Do you know if anyone has
studied realizability triposes by looking at their associated
allegories/bicategories of relations, and/or compared (in that context or
another) the category-of-pers construction of realizability toposes with
the exact-completion method?


FL

> Best,
>   Jim Lipton
>
> On Fri, Dec 10, 2010 at 12:12 PM, <flawler@scss.tcd.ie> wrote:
>
>> Hello all,
>>
>> I've been trying to understand the different constructions of
>> realizability toposes, namely the category of pers in a tripos and the
>> exact completion of a category of partitioned assemblies, and how they
>> are
>> related.
>>
>> I think that the category-of-pers construction can be recast as follows:
>> considering a tripos as a monoidal fibration, take the corresponding
>> bicategory of relations (as in e.g. [1]), split the symmetric
>> idempotents,
>> and take the bicategory of maps, which turns out to be equivalent to a
>> 1-category that in fact is a topos.
>>
>> This can be done in two stages, by splitting first the coreflexives and
>> then the equivalence relations.  The result of the first step (at least
>> in
>> the case of the effective tripos) is the category of assemblies.  The
>> latter is also the regular completion of the category of partitioned
>> assemblies.
>>
>> If this is right, then the effective topos is the category of maps in
>> the
>> symmetric-idempotent splitting of two (non-equivalent) bicategories: the
>> bicategory arising from the effective tripos, and the (local preorder
>> reflection of the) bicategory of spans of partitioned assemblies.
>>
>> I would like to be able to express all of this, and to compare the two
>> kinds of construction further, in the language of cartesian
>> bicategories,
>> which seems like the natural context (especially if one doesn't want to
>> assume that everything is locally preordered).  The closest thing in the
>> literature that I'm aware of is [2], but that paper predates a lot of
>> recent results on cartesian bicategories and on realizability.
>>
>> So my questions are these: does this make sense, or have I made some
>> kind
>> of silly mistake?  Has there been any work on understanding and
>> comparing
>> the usual realizability constructions in this sort of context?
>>
>> Thanks for your attention so far, and thanks in advance for any replies.
>>
>>
>> Finn Lawler
>>
>>
>> References:
>>
>> [1] Mike Shulman, `Framed bicategories and monoidal fibrations', TAC
>> 20(18), 2008
>>
>> [2] Duško Pavlovic, `Maps II: Chasing diagrams in categorical proof
>> theory', Logic Journal of the IGPL, 4(2), 1996
>>
>>
>>
>>
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>

--
Finn Lawler



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