size_question_bisbis

size_question_bisbis

[Eduardo Dubuc]

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 ?

*******************************


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