Two topos questions
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
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
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.
participants (3)
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Peter Arndt -
Prof. Peter Johnstone -
wlawvere＠buffalo.edu