Re: Two topos questions

Re: Two topos questions

[wlawvere]

What I was talking about 15 Jan 1997 was
(not hoping for an axiom of infinity without assuming one, but)

the fact that most of the mathematical uses of the rig N of natural
numbers do not work in a topos, if one interprets that rig to mean the one
characterized by Dedekind recursion.

1. starting with characteristic functions of subobjects, then adding and
multiplying them for various combinatorial calculations
2. applying the least number principle
3. measuring the fiber dimension of a bundle of linear spaces

all require the inf-completion of N, also known as the semicontinuous
natural numbers. It contains the truth-value object omega and is contained
in the semicontinuous reals (themselves indispensible for norming internal
Banach spaces, and constructible simply as one-sided Dedekind cuts).

Yet another way to picture these objects in the case of a Grothendieck
topos E is to consider the sheaf of germs of continuous maps from E to the
appropriate locale : the order topology (not the discrete one) on N, the
order topology (not the interval topology) on nonnegative reals.

Is any more known now as opposed to 9 years ago about the mathematical
applications of finiteness to variable and cohesive sets ? The fact that
K-finiteness is appropriate for some applications and that its theory
resembles the classical theory for constant discrete sets should not
distract us from the achievements of geometers in using coherence,
Notherianness,etc., nor from the fact that our "logic" should serve to
partly guide the learning of also those developments of thought.

Re: Two topos questions

[Prof. Peter Johnstone]

On Wed, 2 Nov 2005, Peter Arndt wrote:

> Hi, category theorists,
>  1. In a message to the categories list from 15. jan.1997 (that message can
> be seen at http://www.mta.ca/~cat-dist/catlist/1999/finite-topos) Lawvere
> talks about "the ... internal topos ... which parametrizes the decidable
> K-finites". Does anyone know what exactly is that internal topos? Is there
> some morphism that can be seen as the indexed family of decidable K-finites
> (just like the generic cardinal "is" the indexed family of finite cardinals
> and can be used to construct the full internal subcategory of finite
> cardinals)?

I can't remember exactly what Bill was talking about in that posting.
However, there is no hope of `parametrizing' decidable K-finite objects
by an internal category, unless the ambient topos has a natural number
object (cf. the remarks on pp. 1058-9 of "Sketches of an Elephant"), and
if it does the decidable K-finites are exactly the objects locally
isomorphic to finite cardinals. So I suspect that he was referring to
the internal category of finite cardinals.

>  2. An object Y of a topos is said to have locally a property P if there is
> an object Z with global support such that Z*(Y) has the property P. For the
> topos of sheaves on a T1-space X (and a property P stable under pullback
> along subterminals), I convinced myself that this implies the existence of a
> covering of X, such that P holds on the restriction of Y to each open set of
> the covering. Can this also be proved for schemes or other classes of
> topological spaces, maybe with additional conditions on P?

Yes, of course -- this is exactly the geometric intuition behind this
use of "locally". One needs to assume that P is stable under arbitrary
pullback (which will certainly be the case if it's expressible in the
internal language of a topos). Then, in any topos generated by
subterminals (in particular, in any topos of sheaves on a space),
every cover Z -->> 1 is dominated by one of the form
\coprod_i U_i -->> 1, where the U_i are a family of subterminals
covering 1 in the classical sense. So P holds locally for Y iff it
holds for the restriction of Y to each member of some cover in
the classical sense.

Peter Johnstone

Two topos questions

[Peter Arndt]

Hi, category theorists,
 1. In a message to the categories list from 15. jan.1997 (that message can
be seen at http://www.mta.ca/~cat-dist/catlist/1999/finite-topos) Lawvere
talks about "the ... internal topos ... which parametrizes the decidable
K-finites". Does anyone know what exactly is that internal topos? Is there
some morphism that can be seen as the indexed family of decidable K-finites
(just like the generic cardinal "is" the indexed family of finite cardinals
and can be used to construct the full internal subcategory of finite
cardinals)?
 2. An object Y of a topos is said to have locally a property P if there is
an object Z with global support such that Z*(Y) has the property P. For the
topos of sheaves on a T1-space X (and a property P stable under pullback
along subterminals), I convinced myself that this implies the existence of a
covering of X, such that P holds on the restriction of Y to each open set of
the covering. Can this also be proved for schemes or other classes of
topological spaces, maybe with additional conditions on P?
 Thanks a lot!
 Peter