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