Subject: categories: Two topos questions

From: Peter Arndt <[email-masked]>
Date: Wed, 2 Nov 2005 03:53:43 -0300

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

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From: "Prof. Peter Johnstone" <[email-masked]>
Date: Wed, 2 Nov 2005 21:22:46 +0000 (GMT)

> and can be used to construct the full internal subcategory of finite

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.

> 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