Re: size_question_encore

[Michael Shulman <mshulman@ucsd.edu>]

Even in a category of sets, I don't see why choice is necessary in
order to complete a small subcategory under finite limits and obtain a
small subcategory.  It seems to me that what is needed is rather the
axiom of collection, which implies that we can find some *set* of
objects containing *at least one* limit for every finite diagram in
the original small subcategory; and then we can iterate countably many
times to obtain a small category which contains at least one limit for
any finite diagram therein.  There is of course no canonical result,
and the various results obtained will not necessarily be strongly
equivalent, but it seems to me that they should all be weakly
equivalent.

And it also seems to me that the same approach should work internal to
any topos.  Collection is true internally to any topos (essentially by
the internal definition of "indexed family"), so it should still be
possible to enlarge a small internal site of definition to one that
has finite limits.  Unless there is some other subtlety that I'm not
seeing.

Mike

On Tue, Jul 5, 2011 at 4:29 PM, Eduardo Dubuc <edubuc@dm.uba.ar> wrote:
> 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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