Re: Logical consequences of descent theory

[Marek Zawadowski <zawado@mimuw.edu.pl>]

The model-theoretic meaning of the descent theorem is in section 11 of

Marek Zawadowski, Descent and duality. Ann. Pure Appl. Logic 71(2), s.
131-188, 1995.

In few words, it says that the formulas of the 'larger' theory invariant
under the
homomorphisms of the 'smaller' theory 'comes' from that smaller theory.
Some other relevant papers to this approach

Michael Makkai, Duality and Definability in First Order Logic, Mem. Of
AMS, no 503, (1993).

Marek Zawadowski, Lax descent theorems for left exact categories.
Dissertationes Math. 346,  (1995).

David Ballard, William Boshuck,Definability and Descent. Journal of
Symbolic Logic 63 (2):372-378, (1998).

Best regards,
Marek

Bas Spitters pisze:
> Dear Dusko,
>
> Thanks. This is interesting, did the Hyland/Moerdijk manuscript you
> cite ever appear?
>
> However, maybe I should have phrased my question as:
> Barr's theorem has an interesting logical corollary.
> This corollary has been used (impressively) by people like Mulvey,
> Vermeulen and Wraith to obtain mathematical results.
>
> I understood that it was suggested that a similar use has been made of
> descent theory.
> Maybe I misunderstood.
>
> Best,
>
> Bas
>
> On Wed, Feb 3, 2010 at 9:02 PM, Dusko Pavlovic <dusko@kestrel.edu> wrote:
>
>> On Feb 3, 2010, at 6:57 AM, Bas Spitters wrote:
>>
>>
>>> A number of people have suggested that descent theory has been/ can be
>>> used to obtain logical results.
>>>
>> [snip]
>>
>>> Any suggestions or pointers about the logical interpretation of
>>> descent theory would be appreciated.
>>>
>> long long time ago there was a paper about the logical meaning of descent
>> with the beck-chevalley condition:
>>
>> @inproceedings{PavlovicD:interpolation,
>>  author =       "Dusko Pavlovic",
>>  title =        "Categorical interpolation: descent and the
>>                  Beck-Chevalley condition without direct images",
>>  booktitle =    "Category Theory, Proceedings, Como 1990",
>>  editor =       "A.~Carboni et al.",
>>  publisher =    "Springer Verlag",
>>  series =    LNM,
>>  volume =    "1488",
>>  pages =        "306--326",
>>  year =         "1991"
>> }
>>
>> more interestingly, one can also go back, and work out the exact logical
>> conditions for descent, which are weaker than the beck-chevalley.
>>
>> -- dusko
>>
>>
>>
>
>
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