Subject: question on finiteness in toposes

From: categories <[email-masked]>
Date: Fri, 10 Jan 1997 12:32:35 -0400 (AST)

Date: Fri, 10 Jan 1997 12:57:02 MEZ
From: Thomas Streicher <[email-masked]>

One knows that for any topos E that the full subcategory of decidable K-finite 
objects forms a topos itself with 2 = 1+1 as subobject classifier.
It is also said that E_kf, the full subcat of E on K-finite objects need not
form a topos. That's what I could find out from PTJ's Topos Theory. 
The counterexample given there is E = Set^2 (where 2 = 0 -> 1). The K-finite
objects in Set^2 are the surjective maps between finite sets. It is clear that
E_kf is not closed under equalisers taken in E (!). Nevertheless, I think that
E_kf itself does have equalisers : if f,g : X -> Y then take the equaliser 
e_0 of f_0 and g_0 and take the epi-mono-factorisation of x o e_0 :

     E_0 >---> X_0
      |         |
      |  epi    | x        where  X = X_0 -> X_1
      V         V
     E_1 >---> X_1 

this clearly demonstrates that the inclusion  E_kf >--> E does not preserve 
equalisers BUT it does not show that E_kf is not a topos.
I would be interested in a reference or example where E_kf really is not a 
topos. Maybe, E = Set^2 alraedy works but it must have another defect than 
not being clossed under subobjects w.r.t. E because the decidable K-finite
objects have this "defect" as well.
Thomas Streicher

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From: categories <[email-masked]>
Date: Sat, 11 Jan 1997 13:15:36 -0400 (AST)

Date: Sat, 11 Jan 1997 09:05:39 -0500 (EST)
From: Peter Freyd <[email-masked]>

Let me expand. If one bores into just why  Set^2_kf  can't be
boolean and looks for a minimal example of its non-booleaness
one inevitably lands on the object  2 -> 1. At first blush
its lattice of subobjects does look boolean. Until one notices
that there's a monomorphism from  2 -> 2  to  2 -> 1 (where
2 -> 2  is the identity map).

Having noticed that, one has a quicker proof that it's not a
topos: not every mono-epi is an equalizer.

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From: categories <[email-masked]>
Date: Sat, 11 Jan 1997 13:14:47 -0400 (AST)

Date: Sat, 11 Jan 1997 08:38:43 -0500 (EST)
From: Peter Freyd <[email-masked]>

  The answer that first occures me for Thomas Streicher's question is
  that in  Set^2_kf  the terminator generates, hence if it were a topos 
  it would have to be a boolean topos. Which it clearly isn't.

  Thomas wrote:

     E_0 >---> X_0
      |         |

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From: categories <[email-masked]>
Date: Sun, 12 Jan 1997 16:42:23 -0400 (AST)

Date: Sun, 12 Jan 97 01:31 EST
From: Fred E J Linton <[email-masked]>

Supplementing Peter's answer to Streicher's K-finiteness question,
I recall Prop. 7.4 on p. 97 of SLNM #753, which states, for presheaf topoi
E = (C^op, Sets), that, with  E_Kf  the full subcategory of K-finite E-objects:
 
E_Kf  is balanced
iff
it's a topos
iff
each K-finite is decidable
iff
C is a "2-way" category
iff
... .

Streicher's >--> sure isn't 2-way, hence ... .

The rest of that 20 year old report on my student Acun~a's thesis with me
is also fun.

-- Fred

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From: categories <[email-masked]>
Date: Mon, 13 Jan 1997 10:26:43 -0400 (AST)

Date: Sun, 12 Jan 1997 16:53:57 -0500 (EST)
From: F William Lawvere <[email-masked]>

Now that the question of finiteness as been reactivated here,
may I bring up again the following question  ?
 
What concept of finiteness is appropriate for those important
mathematical applications in topology for which K/S doesn't
seem right ? (For example the equalizer closure of K/S or...??)
Especially, a suitably "finite" module should be a vector bundle
or a FAC in the sense of Serre so that our simplified topos theory
could apply more directly to those things it should.
Bill L

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From: categories <[email-masked]>
Date: Tue, 14 Jan 1997 20:14:14 -0400 (AST)

Date: Tue, 14 Jan 1997 10:52:38 -0500 (EST)
From: F William Lawvere <[email-masked]>

Sorry, I used K/S for an abbreviation of what was called
Kuratowski until someone pointed out that it was due to
Sierpinski :an object whose mark belongs to the smallest
sub-semilattice of its power set which contains the
singleton map, or in case there is an NNO an object
which in a suitable sense is locally enumerable by
the segment under a section of the NNO .

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, still
the theory of it in the last 25 years of topos theory seems
to mainly be justified by formal analogy and/or independence
relative to abstract set theory (=topos with choice).
However there are important uses of "finiteness" in
algebraic geometry and differential topology (where topos theory
after all started)   :
Consider a ringed topos E,R . For example, the sheaves on an
algebraic variety  or on a Cinfty manifold. Within the abelian
category of R-modules in E, we need to single out two important
subcategories
FAC (Serre 1955)=coherent sheaves..these tend to be an abelian
subcategory and tend to vary covariantly as one E,R is mapped to
another E',R' (thus give rise to an extensive K-homology)
and vector bundles , which one thinks of as a finite-dimensional
vector space varying smoothly over the base space of E ,so
they cry out for internalization ; in algebraic geometry these
are identified with locally FINITELY free R-modules... they
vary contravariantly with E,R  (so give rise to K-cohomolgy
rigs which act on the FACs,ie intensives acting as densities on
the extensives; with further conditions on E,R one can at the
level of the riNgs generated by these rigs define a sort of
Radon/Nikodym derivative via an alterating sum of Tors , but
in general the covariant abelian category FAC and the
contravariant tensored category Vect are distinct...The
"derived category" of E,R (now allegedly replacing homological
algebra in complex analysis and C*-algebra theory)  should
be the derived category of one of these two linear categories
(here I mean dc in the linear sense..nonlinear "derived categories"
are more like the stable homotopy of E))

Already the intuitionists speculated about (in effect) subobjects
of K/S objects, and  it seems we need something of the sort
perhaps a category of finites closed under subquotient in
order to define the notion of eg finitely-generated R-module
in a way which not merely mimics abstract set theory but actually
captures the vector bundles .

Perhaps it will be easier if E itself satisfies a noetherian
condition.

It would be best if the desired content could be entirely int-
ernalized to E,R but perhaps it is really relative to a base
S,K..but perhaps without restriction on S ??

I hope this clarifies the problem.
Sincerely
Bill

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From: categories <[email-masked]>
Date: Wed, 15 Jan 1997 10:33:21 -0400 (AST)

Date: Wed, 15 Jan 97 10:19 GMT
From: Dr. P.T. Johnstone <[email-masked]>

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

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From: categories <[email-masked]>
Date: Wed, 15 Jan 1997 21:25:06 -0400 (AST)

Date: Wed, 15 Jan 1997 15:22:00 -0500 (EST)
From: F William Lawvere <[email-masked]>

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.