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