RE: question on finiteness in toposes

[categories <cat-dist@mta.ca>]

Date: Wed, 15 Jan 1997 15:22:00 -0500 (EST)
From: F William Lawvere <wlawvere@ACSU.Buffalo.EDU>

Concerning Peter Johnstone's clarification: Of course I didn't
mean that the object classifier could be constructed without
an internal parameterizer for the finite objects in the base S
.... but what exactly are the finite objects ?
  While the classifier as a topos is determined by the 2-category of
bounded S-toposes , the site for it isn't. I was under the impression 
that an internal category parameterizing the objects which are both
K-finite and separable(=decidable) could be used (while internal
presheaves on "all" K=finites would presumably be much bigger..what
does IT classify ?) Anyway my point was that at any rate no further
extension of the notion of finiteness is needed for classifying in
that sense the objects or the group objects in S-toposes, whereas by
contrast it seems that to give the mathematically correct notion of
"vector space for which there exists a finite basis" does need such an
extension.

In any topos, a subobject of a nonnon sheaf is always separable ;
when is the converse true  ? 

Perhaps there is an internal topos object V which is largest with
respect to being fully embedded in the given topos E while at the 
same time having A as its subtopos of internal nonnon sheaves. Here
by A is meant the Boolean internal topos mentioned above which
parameterizes the separable K-finites of E (Fred recalled Acunya's
work showing among other things that it is Boolean) and to say
that V "is" fully embedded in E has sense for any internal category
with a terminal object  , namely we require that the canonical
parametrized (="indexed") functor from V to E is an equivalence
E(X,V)--> E/X  for each X. The latter functor is defined by merely
pulling back the fibration 1/V--> V of pointed objects in V.
When the answer to the above question is affirmative, Johnstone's
locally separable reflection Vsubqd will consist of subquotients and
the K-finites may fit in . It seems that the inclusion of A in V
will preserve sums but only certain epis.
The idea is that V can't be too large since the inverse to the
inclusion will enrich it in A.
 
On Wed, 15 Jan 1997, categories wrote:

> Date: Wed, 15 Jan 97 10:19 GMT
> From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk>
> 
> Not an answer to Bill's question (which I agree is an important one),
> but a minor correction. Bill wrote:
> 
> While the K/S definition is right for the construction of
> the object classifier over an arbitrary base topos (as Gavin
> showed) and hence for classifiers for various kinds of
> finitary algebras over an arbitrary base topos,
> 
> It isn't, and he didn't. Gavin used finite cardinals to construct
> the object classifier over an arbitrary base topos with NNO (and I
> subsequently extended the construction to finitary algebraic
> theories), but it doesn't work over a topos without NNO (and in
> particular it can't be made to work using K-finiteness). Andreas
> Blass showed that the existence of an object classifier for toposes
> over E implies that E has a NNO.
> 
> Incidentally, I think it is correct to give credit to Kuratowski for
> the notion of K-finiteness. It's true that Sierpinski's paper was
> earlier, but his definition was a "global" one (i.e. he defined the
> class of all finite sets as the sub-semilattice of the universe
> generated by he singletons), whereas Kuratowski made the crucial
> observation that the finiteness of a particular set X can be determined
> locally (i.e. within the power-set of X), without which the notion
> could never have been imported into topos theory.
> 
> Peter Johnstone
>