Two questions on the real numbers

[Alex Simpson <als@inf.ed.ac.uk>]

While the topic is still hot, I'd like to post two more questions
concerning the various real number objects.

Just to recap, in any elementary topos with nno N we have the Cauchy
reals R_C, the Dedekind reals R_D, and also the "Cauchy-completion
(w.r.t. sequences) of R_C in R_D", as discussed thoroughly in previous
postings. Let's call this latter object R_E.

All three objects are logically defined, and hence preserved by
logical functors between toposes. Martin Escardo and I proved
that R_E is also preserved by inverse image functors of
essential geometric morphisms. (Actually, we proved this for
the closed unit interval I_E in R_E. The result for R_E follows
by exibiting R_E as a coequalizer of two well-chosen maps from
N x I_E to itself.)

QUESTION 1. What preservation results are available for
  R_C and R_D?

Surely something is known about this. I can't, at the moment,
see any reason for them to be preserved by inverse image
functors of arbitrary essential geometric morphisms.


Interesting mathematical differences between the objects arise
when one considers algebraic closure properties of the
complex numbers C_C, C_D and C_E associated with R_C,
R_D and R_E respectively.

In Fourman and Hyland, "Sheaf models for analysis" (Springer
LNM 753, 1979) it is shown that the Dedekind complex numbers
C_D are not in general algebraically closed. This fails
in the strong sense that, in certain toposes, one can actually
exhibit a polynomial that has no root. Thus the fundamental
theorem of algebra does not hold in general for C_D.

On the other hand, in Ruitenberg, "Constructing roots of polynomials
over the complex numbers" (in "Computational aspects of Lie group
representations and related topics", CWI Tract 84, Amsterdam, 1990),
it is shown that the fundamental theorem of algebra does hold
for C_C in any topos. (I first heard about this paper from Fred
Richman. I have never seen it. I have only read its math review,
no.92g:03085.)

The conflicting picture above, leads naturally to:

QUESTION 2. Does the fundamental theorem of algebra hold for C_E?


I would be interested to hear any ideas at all related to these
questions.

Alex Simpson

Alex Simpson, LFCS, Division of Informatics, Univ. of Edinburgh
Email: Alex.Simpson@ed.ac.uk           Tel: +44 (0)131 650 5113
Web: http://www.dcs.ed.ac.uk/home/als  Fax: +44 (0)131 667 7209