size_question

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

This is a naive question on non naive foundations.

Consider the inclusion S_f C S  of finite sets in sets.

Is the category S_f closed under finite limits and at the same time small ?

For example, there are a proper class of singletons, all finite. Thus a
proper class of empty limits.

Question, which is the small category of finite sets ?, which are its
objects ?.

A small site with finite limits for a topos would not be closed under
finite limits ?

etc etc

But, more basic is the question above: How do you define the small
category of finite sets ?

Or only there are many small categories of finite sets ?

You can not define a finite limit as being any universal cone because
then you get a large category.

Then how do you determine a small category with finite limits without
choosing (vade retro !!) some of them. And if you choose, which ones ?

The esqueleton is small but a different question !!

e.d.


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