On Fri, 7 Dec 2001 S.J.Vickers@open.ac.uk wrote:
Does any one know the answers to these questions?
1. Is trigonometry valid in toposes? (I'll be astonished if it isn't.) 2. Does a polynomial over the complex field C have only finitely many roots?
More precisely:
1. Over any topos with nno, let R be the locale of "formal reals", i.e. the classifier for the geometric theory of Dedekind sections.
Do sin, cos, arctan, etc. : R -> R exist and satisfy the expected properties? Are there general results (e.g. on power series) that say Yes, of course they do?
The space of Dedekind reals is Cauchy-complete, so any convergent power series such as sin or cos defines an endomorphism of it. Moreover, provided (as in this case) we can calculate a "modulus of convergense" for the power series explicitly from a bound for x, it's easy to see that the construction x |--> sin x commutes with inverse image functors, so it must be induced by an endomorphism of the classifying topos (that is, of the locale of formal reals).
2. Consider the space S of square roots of the generic complex number. Working over C, it is the locale corresponding to the squaring map s: C -> C, z |-> z^2. The fibre over w is the space of square roots of w.
s is not a local homeomorphism, so S is not a discrete locale. Hence we can't say S is even a set, let alone a finite set in any of the known senses. I don't believe its discretization pt(S) is Kuratowski finite either. If I've calculated it correctly, it is S except for having an empty stalk over zero (oops!), and there is no neighbourhood of zero on which an enumeration can be given of all the elements of pt(S).
On the other hand, S is a Stone locale - one can easily construct the sheaf of Boolean algebras that is its lattice of compact opens. That sheaf of Boolean algebras is not Kuratowski finite, nor even, it seems to me, a subsheaf of a Kuratowski finite sheaf.
So is there any sense at all in which S is finite?
That's a good question. I've never thought about notionss of finiteness for non-discrete locales (someone should!). For the set of points of S, I believe it should be what Peter Freyd called "R-finite" ("R" for "Russell"): intuitively, this means that there is a bound on the size of its K-finite subsets. (However, I don't have a proof of this.) R-finiteness is quite a lot weaker than \tilde{K}-finiteness (being locally a subobject of a K-finite object), but it's still a reasonably well-behaved notion of finiteness (e.g. it is preserved by functors which preserve all finite limits and colimits). Peter Johnstone