I got a private answer that made me realize I was not clear in the way I put my questions. I send now this mail with my answers, which I hope will clarify the situation. e.d. ****************** Perhaps I was not clear in my question ? It is not belived that finite sets is a small topos with finite limits and all the rest exept the axiom of infinity ? Which category of finite sets ? Of course I know they are all equivalent, but there is not a canonical small one. The answer is not, but people behave as if it were true. More comments below:
I'm replying "in private" to minimize embarrassment.
Consider the inclusion S_f C S of finite sets in sets.
Is the category S_f closed under finite limits
If you mean to understand, as do I, that S_f stands for the full subcategory of the category S of sets whose objects are the sets that happen to be finite, then: YES, of course; ...
... and at the same time small ?
... and NO, obviously not.
For example, there are a proper class of singletons, all finite. Thus a proper class of empty limits.
Quite so. Do you find that objectionable?
Question, which is the small category of finite sets ?, which are its objects ?.
I don't understand either the question or its preassumptions. Why should there be a unique ("the") category of finite sets?
Well, there is a unique category of finite sets, but it is not small, but people works as it it were. Take Joyal's theory of species.
The category S_f of finite sets discussed above isn't small. But it has lots of equivalent, small subcategories, including skeletal ones, the best known of which is the full subcategory of S whose objects are the finite cardinals. Which of these (if any) is closed under finite limits is an imponderable.
Yes, what happens then when people works with a small category of finite sets and with finite limits ?
A small site with finite limits for a topos would not be closed under finite limits ?
Closed in relation to what ambient setting?
The topos.
etc etc
But, more basic is the question above: How do you define the small category of finite sets ?
I *don't* define "the" small category of finite sets. If I need *a* skeletal version of the category of finite sets, though, I settle for that full subcategory of finite ordinals.
What if I need a small category of finite sets with finite limits. Well, using choice I can produce one starting with the finite ordinals (or cardinals). Choosse a limit for each finite diagram, and keep doing this a dennumerable amount of times.
Or only there are many small categories of finite sets ?
So I would think :-) .
Yes, me too, so this is presisely why I ask the question.
As to what follows here, I either have no answer for it, or do not even understand the content or relevance of it:
Well, I did not explain myself clearly.
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 !!
So if you'd like to amplify a bit, or to explain somewhat, I'd be grateful.
PS: I'm suddenly aware that June means WINTER in Argentina, so I hope
your
questions are not merely symptomatic of a winter "brain-freeze"
Well, it is not "brain freeze" yet ... or it is ? ******************************* [For admin and other information see: http://www.mta.ca/~cat-dist/ ]