size_question_reloaded

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

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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