size_question_encore

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

I have now clarified (to myself at least) that there is no canonical
small category of finite sets, but a plethora of them. The canonical one
is large. With choice, they are all equivalent, without choice not.

When you work with an arbitrary base topos (assume grothendieck) "as if
it were Sets" this may arise problems as they are beautifully
illustrated in Steven Vickers mail.

In Joyal-Tierney galois theory (memoirs AMS 309) page 60, they say S_f
to be the topos of (cardinal) finite sets, which is an "internal
category" since then they take the exponential S^S_f. Now, in between
parenthesis you see the word "cardinal", which seems to indicate to
which category of finite sets (among all the NON equivalent ones) they
are referring to.

Now, it is well known the meaning of "cardinal" of a topos ?.
I imagine there are precise definitions, but I need a reference.

Now, it is often assumed that any small set of generators determine a
small set of generators with finite limits. As before, there is no
canonical small finite limit closure, thus without choice (you have to
choose one limit cone for each finite limit diagram), there is no such a
thing as "the" small finite limit closure.

Working with an arbitrary base topos, small means internal, thus without
choice it is not clear that a set of generators can be enlarged to have
a set of generators with finite limits (not even with a terminal
object). Unless you add to the topos structure (say in the hypothesis of
Giraud's Theorem) the data of canonical finite limits.

For example, in Johnstone book (the first, not the elephant) in page 18
Corollary 0.46 when he proves that there exists a site of definition
with finite limits, in the proof, it appears (between parenthesis) the
word "canonical" with no reference to its meaning. Without that word,
the corollary is false, unless you use choice. With that word, the
corollary is ambiguous, since there is no explanation for the technical
meaning of "canonical". For example, in theorem 0.45 (of which 0.46 is a
corollary), the word does not appear. A topos, is not supposed to have
canonical (whatever this means) finite limits.

e.d.


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