Date: Fri, 10 Jan 1997 12:33:03 -0400 (AST)
Subject: question on finiteness in toposes
Date: Fri, 10 Jan 1997 12:57:02 MEZ
From: Thomas Streicher
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
Date: Sat, 11 Jan 1997 13:15:27 -0400 (AST)
Subject: Re: question on finiteness in toposes
Date: Sat, 11 Jan 1997 08:38:43 -0500 (EST)
From: Peter Freyd
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:
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
Date: Sat, 11 Jan 1997 13:16:06 -0400 (AST)
Subject: Re: question on finiteness in toposes
Date: Sat, 11 Jan 1997 09:05:39 -0500 (EST)
From: Peter Freyd
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.
Date: Sun, 12 Jan 1997 16:42:58 -0400 (AST)
Subject: RE: question on finiteness in toposes
Date: Sun, 12 Jan 97 01:31 EST
From: Fred E J Linton <0004142427@mcimail.com>
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
Date: Mon, 13 Jan 1997 10:27:19 -0400 (AST)
Subject: RE: question on finiteness in toposes
Date: Sun, 12 Jan 1997 16:53:57 -0500 (EST)
From: F William Lawvere
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
Date: Tue, 14 Jan 1997 20:15:08 -0400 (AST)
Subject: RE: question on finiteness in toposes
Date: Tue, 14 Jan 1997 10:52:38 -0500 (EST)
From: F William Lawvere
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
Date: Wed, 15 Jan 1997 10:33:30 -0400 (AST)
Subject: RE: question on finiteness in toposes
Date: Wed, 15 Jan 97 10:19 GMT
From: Dr. P.T. Johnstone
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
Date: Wed, 15 Jan 1997 21:25:46 -0400 (AST)
Subject: RE: question on finiteness in toposes
Date: Wed, 15 Jan 1997 15:22:00 -0500 (EST)
From: F William Lawvere
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
>
> 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
Date: Fri, 17 Jan 1997 16:04:52 -0400 (AST)
Subject: RE: Finiteness in Toposes
Date: Fri, 17 Jan 1997 12:50:13 -0500 (EST)
From: F William Lawvere
Re: Finiteness in Toposes Jan 17 1997
This concerns the possibility , mentioned in my previous message, of two
internal toposes of finite objects.
The conjecture that there are two natural internal categories of
finite objects is partly supported by the fact that there are two natural
natural-numbers objects, the usual one N that parameterizes compositional
iteration and another semicontinuous one L with the following features:
0) It is a rig, so receives a homomorphism from N and its
elementary arithmetic starts out looking very similar.
1) But unlike N it has a least-number-property in the sense that it is
inf-complete and better.
2) It can be constructed internally using truth-valued sheaves on N.
3) Hence it also contains a map from (big) omega, which permits (unlike
N) the use of the standard method in finite combinatorics where (for
example) a binary relation is considered as a matrix which is valued (not
only in a rig where 1+1 = 1, but instead) in a rig in which natural
numbers are distinct; the resulting generalized characteristic functions
are added, multiplied, infed etc. according to the usual methods of
arithmetic and analysis and then translated back into the combinatorics of
the original finite structures. Of course, in each case one hopes that
the answer to a combinatorial problem might turn out decidable, but that
shouldnt require us to stay in the bounds of two-valued subsets in the
course of a construction.
4) This internally-defined order-complete rig in E has also an external
characterization if E is an S-based topos, namely it is the sheaf of germs
of S-geometrical morphisms from E to the topos often called S-sets
-through-time (I dont think that depends on any presumption that the N in
S ,used to parameterize the transitions through time, coincides with its
completion in S). In localic or open set terms, there is in S a (T sub
zero) space whose points are N, but whose open sets have the usual order
on N as their specialization order; continuous functions from any space E
to this space are called semi-continuous and there is in E a sheaf of
them.
5) The application to the variable linear algebra over algebraic or
complex-analytic spaces needs L too, because dimension of a vector space
is a semi-continuous function. More precisely, if A is a good module in
a ringed topos E, R then for each X and E there should be a map X--> L
which is the fiber-wise dimension of X*A. The basic case is perhaps that
where E,R is an algebraic affine scheme, and the conceptual problem is to
get at what sort of sets contained in A this dimension function is
counting (or bounding). One should not expect that equality of dimension
will imply isomorphism.
This object L has been discussed for 25 years, but I dont know if
anyone published the working-out of its properties and role.
Bill
Date: Wed, 22 Jan 1997 14:41:45 -0400 (AST)
Subject: RE: Finiteness in Toposes
Date: Wed, 22 Jan 1997 18:18:53 +0000
From: Steve Vickers
>Date: Fri, 17 Jan 1997 12:50:13 -0500 (EST)
>From: F William Lawvere
>
>Re: Finiteness in Toposes Jan 17 1997
>
>This concerns the possibility , mentioned in my previous message, of two
>internal toposes of finite objects.
>
> The conjecture that there are two natural internal categories of
>finite objects is partly supported by the fact that there are two natural
>natural-numbers objects, the usual one N that parameterizes compositional
>iteration and another semicontinuous one L with the following features:
>
>...
>
> This object L has been discussed for 25 years, but I dont know if
>anyone published the working-out of its properties and role.
Am I right in thinking this to be Idl N, the ideal completion of the
natural numbers (with their usual order)?
I conjecture that this is a suitable value domain for the ranks of matrices
over localic fields such as the reals: rank^-1{n} is not open, but
rank^-1{n, n+1, n+2, ...} is. Then rank A is the set of natural numbers n
such that we can find enough apartnesses to prove linear independence of n
rows of A, and this is an ideal of N - the definition also smoothly
incorporates infinite matrices.
(Perhaps this is just one of the things that have have been discussed for
25 years and I'm reiventing it.)
Anyway, I have investigated Idl N as a fixpoint object (in the sense of
Crole and Pitts) in the category of Grothendieck toposes (modulo
2-categorical niceties that I didn't investigate too closely) in a paper
"Topical Categories of Domains".
Steve Vickers.
Date: Thu, 23 Jan 1997 14:50:30 -0400 (AST)
Subject: finiteness
Date: Thu, 23 Jan 1997 19:43:40 EST
From: carboni@vmimat.mat.unimi.it
Regarding the last message of Bill on finiteness and the answer of Vickers, I
would like to point out that the 25 years old reference of Bill should be the
one at the bottom of page 14 of lesson 3 of 1972 Perugia Notes. In other words,the L he is suggesting should be the internalization of the following
description: L consists of those ideals T : N---->Omega such that for all n
Tn = Inf{Tm | m > n}.
The precise meaning of this definition is explained in the given reference, as
well as in Bill's messages. I hope that this is correct.
Aurelio Carboni
Date: Fri, 24 Jan 1997 10:56:07 -0400 (AST)
Subject: Re: finiteness
Date: Fri, 24 Jan 1997 10:54:59 +0000
From: Steve Vickers
>From: carboni@vmimat.mat.unimi.it
>
>... L consists of those ideals T : N---->Omega such that for all n
>Tn = Inf{Tm | m > n}.
I understand this as saying that n is in the ideal iff every greater m is
in the ideal (but I think the inequality m > n has to be non-strict to make
sense of this). Hence it's really a filter of N.
If that's correct, then my suggestion was wrong. L would be not Idl N, but
Idl(N^op). That makes sense regarding dimensions, for if a real vector
space is finitely presented using an mxn matrix A (presenting R^n/Im A)
then its dimension is n-rank(A), so if rank(A) is in Idl(N), the dimension
should be in Idl(N^op).
(By the way, what's a full reference for the "Perugia Notes"?)
Steve.
Date: Mon, 27 Jan 1997 13:15:55 -0400 (AST)
Subject: finiteness
Date: Sun, 26 Jan 1997 18:16:28 EST
From: carboni@vmimat.mat.unimi.it
Regarding my last message on finiteness, I should have said `functors
N---->Omega' instead of `ideals N--->Omega'. I repeat that I `think' that
this is what Bill wanted to say, but I am not sure that I am correct.
As for the reference of Bill's Perugia Notes, they are an internal publication
of Perugia University in 1972 of the lectures given by Bill Lawvere when he
was visiting that University. They should be available there (write to prof.
L. Stramaccia, Dipartimento di Matematica, Universita' di Perugia,
via Vanvitelli 1, 06123 Perugia, Italy, email: stra@gauss.dipmat.unipg.it).
Also, they were quite spread out, so that you should be able to find somebody
nearby you who has them. Other possibilities are asking the author himself and
eventually myself.
Aurelio Carboni.