Logical consequences of descent theory

Logical consequences of descent theory

[Bas Spitters]

A number of people have suggested that descent theory has been/ can be
used to obtain logical results.

I can sort of see the possible connections:
open surjective maps between toposes are effective descent morphisms.
Viewed logically, such a map is a conservative extension preserving
all first-order structure.

Proper surjective maps are also effective descent morphisms.
Consider an occupied locale X (in Paul Taylor's sense).
I.e. X->1 is a proper surjection.
Then we obtain a proper surjection Sh(X)->Sets.
I.e. we conservatively add a generic point of the occupied space.

The inverse image preserves geometric logic, but does it preserve
anything else in general?
This is probably well-known, but I couldn't find it.

Any suggestions or pointers about the logical interpretation of
descent theory would be appreciated.


Thanks,

Bas


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Re: Logical consequences of descent theory

[Dusko Pavlovic]

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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Re: Logical consequences of descent theory

[Bas Spitters]

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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Re: Logical consequences of descent theory

[Marek Zawadowski]

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
>>
>>
>>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


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Re: Logical consequences of descent theory

[William Boshuck]

In connnection with the descent theorem
of Marek for pretoposes, the papers by
Moerdijk and Vermeulen should be mentioned:

Proper maps of toposes, Memoirs of the AMS, Vol 705 (2000).

Proof of a conjecture of Pitts, Volume 143, Number 1 (1999),
pp. 329-338


Descent theory has been a kind of definability
theory from the beginning, I think.  See, e.g.,

Andre Weil, The Field of Definition of a Variety,
American Journal of Mathematics, Vol. 78, No. 3
(Jul., 1956), pp. 509-524.

-wb


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Re: Logical consequences of descent theory

[Colin McLarty]

2010/2/12 William Boshuck <boshuk@math.mcgill.ca>:


Makes a very nice point which I want to underline:

> Descent theory has been a kind of definability
> theory from the beginning, I think.  See, e.g.,
>
> Andre Weil, The Field of Definition of a Variety,
> American Journal of Mathematics, Vol. 78, No. 3
> (Jul., 1956), pp. 509-524.

Colin


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