size_question_reloaded

size_question_reloaded

[Eduardo J. Dubuc]

Thanks to all who wrote something on this question.

It clarified mi ignorance:

There is the category of finite sets, namely, the category of all those 
sets which happen to be finite. No need of more precision. But it is not 
small (or an element of the universe if you like).

If you want small, then there are plenty of them, and anybody can use 
their FAVORITE one. But this is not usually done, it seems that the fact 
that the canonical one is “essentially small” is good enough to dismiss 
all possible problems.

For example, people which consider the presheaf category 
Set^((Set_f)^op)  (object classifier) often do as if Set_f  were 
canonical and small.

Now, if you work with a  Grothendieck base topos “as if it were the 
category of sets”, you are forced to specify which small category of 
finite sets you are using,    or not ?.

Cheers  e.d.


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Re: size_question_reloaded

[William Messing]

Concerning Eduardo's question: Now, if you work with a  Grothendieck 
base topos “as if it were the category of sets”, you are forced to 
specify which small category of finite sets you are using,    or not ?.

No, in the definition of an U-topos, E,  given in SGA 4, Expose IV, § 1, 
it is required that E has an U-small (topologically) generating set of 
objects.  Bu such a set  or, equivalently, full subcategory of E, is not 
part of the data.

Bill Messing


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Re: size_question_reloaded

[Steven Vickers]

Dear Eduardo,

That's right. Specifically, Set_f here has N for its object of objects,
and something more complicated (but geometrically definable) for its object
of morphisms. That way the correct object classifier is defined for any
base topos. (I'm thinking of Grothendieck toposes here, but I guess it
works for any elementary topos with NNO. I even conjecture it does
something useful for arithmetic universes.)

That's a very strong notion of finiteness constructively. It requires not
only Kuratowski finiteness and decidable equality, but even a decidable
total order. Then the category of such finite sets is essentially small,
equivalent to the Set_f I described above. It is the notion of "finite"
needed in finitely presentable algebras, for example in the theorem that
for a finitary algebraic theory T, the T-algebra classifier is
Set^(T-Alg_fp^op), the topos of Set-values functors from the category of
finitely presented algebras. Again, we want T-Alg_fp to be small.

In my paper "Strongly algebraic = SFP (topically)" I was interested in the
situation where, for a geometric theory T, the classifying topos for T is  a
presheaf topos in the form of the topos of Set-valued functors from the
category of finite T-models (and I gave some sufficient conditions for this
to happen). Again, the notion of finite model is this strong notion of
finiteness. However, my main example also involved Kuratowski finite sets,
so the paper discusses in some detail the interplay between the different
notions of finiteness.

Regards,

Steve Vickers.

On Sun, 03 Jul 2011 19:59:16 -0300, "Eduardo J. Dubuc" <edubuc@dm.uba.ar>
wrote:
> ...
> For example, people which consider the presheaf category 
> Set^((Set_f)^op)  (object classifier) often do as if Set_f  were 
> canonical and small.
> 
> Now, if you work with a  Grothendieck base topos “as if it were  the 
> category of sets”, you are forced to specify which small category of 
> finite sets you are using,    or not ?.
> 
> Cheers  e.d.
> 

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