arithmetical and geometric reals in (models of) SDG
Recently I was asking myself what is the relation between the arithmetical (Dedekind) reals in a topos and a ring R satisfying the usual SDG axioms (i.e. at least the Kock-Lawvere) axiom.
From the few things I know about models of SDG it seems to me as if in the usual sheaf models (over a site Loc of `loci') the Dedekind reals form a subring of R = y(\Re) (where \Re is the locus corresponding to the reals). As long as one considers just presheaves that's clear as the Dedekind reals are given by \Delta(\Re). Probably taking sheaves that situation isn't changed dramatically?
So my impression is that in the usual sheaf models of SDG the real line R carries the structure of an algebra over the Dedekind reals. However, I don't see how to construct such an embedding of Dedekind reals into R based only on the usual axioms of SDG. Of course, when given an order on R and R is assumed as a Q-algebra then one has a good candidate for a function from R to R_D sending x \in R to {q \in Q | q.1 \leq x}. But how to define purely logically an embedding of R_D into R remains mysterious (to me). I am fully aware of the fact that my question is a bit `against the strain' of SDG but we know that both kinds of reals do coexist in topos models. But is this coexistence only peaceful or rather more collaborative? I am certain that people must have thought about this but I couldn't find anything at the usual places where to look. So I would be grateful for comments (on the correctness of my above speculations) or pointers to existing literature or folklore. Best, Thomas Streicher
Thomas Streicher <streicher@mathematik.tu-darmstadt.de> writes:
Recently I was asking myself what is the relation between the arithmetical (Dedekind) reals in a topos and a ring R satisfying the usual SDG axioms (i.e. at least the Kock-Lawvere) axiom.
Perhaps it is worth mentioning that in the ring of smooth reals R the sequence a_k = 2^(-k) is Cauchy but has many "limits" because every infinitesimal dx satisfies the condition "dx is the limit of a_k". This shows that R is not Cauchy complete, not because limits of Cauchy sequences are missing but because there are too many. I once thought the above observation implied there can be no isometric embedding of a Cauchy-complete field (e.g. the Dedekind reals) into R, but now I am not convinced anymore. Andrej Bauer Department of Mathematics and Physics University of Ljubljana http://andrej.com
The map from Euler reals to Dedekind reals is not injective (1) The rig of uppercuts in Q serves as value-space for metrics; call it the Dedekind reals for short. (Mapping a ring to the Dedekind reals would only hit two-sided cuts, but that is a separate issue. If Q denotes the nonnegative rationals, then the term "arithmetic reals" would be justified, but for the issue addressed here, Q might as well be "the constant reals" coming from a lower topos). (2) Euler affirmed that a real should be determined as a ratio between infinitesimals. Adopting a rational definition of "ratios", and conservatively interpreting the appropriate space T of infinitesimals as the representing object for the tangent-bundle functor, I call Euler reals the part R of the function-space T^T that preserves the base point. (T is regarded as given as a reflection of physical experience, so not every topos has one. R typically has a unique addition compatible with the obvious multiplication. If we define D as the part of R of square 0, the Kock-Lawvere axiom would require that there exist units of time, i.e., isomorphisms T->D, or equivalently certain non-unique semigroup structures on T itself (in contrast with the canonical multiplication on our R)). (3) Philosophically, the Euler reals serve not only to parameterize motion but also to provide a means to express a cause of motion; the cause operates at each single time, as is reflected in the fact that T has a unique point. By contrast, the Dedekind reals serve to measure, by Q-approximations, the changes resulting from motion between pairs of times. Measuring, like photographing, kills the particular motion; thus the map from Euler reals to Dedekind reals, should not be expected to be injective. That map needs to be understood in any smooth topos of interest,. (a) Of course, measuring can still derive information, perhaps even enough information, about the causes of motion too: we can pass to another moving quantity, e.g. velocity via a speedometer, and then measure that via rational approximation. There is an analogy with algebraic topology: pizero is a very crude measure of a space, it would seem, but as Sammy liked to point out, if you apply an appropriate geometrical endofunctor first, then pizero can deliver lots of useful information.) (b) Any given object in a smooth topos will induce a function presheaf on finite-dimensional varieties; since continuous functions are not usually smooth functions, it is unlikely that the Dedekind reals (even two-sided) will be included in R.) (4) However, an inclusion Q->R of constants is to be expected; it forms one ingredient for constructing the map under discussion. The other ingredient is an ordering on R, inducing in the obvious way the map from R to parts of Q. Several treatments of SDG postulate such an ordering, but it always seems to turn out that the ordering is not anti-symmetric (in particular that any closed interval is closed under the addition of infinitesimals), illustrating the non-injectivity of the map. (In some cases there seem to be ways to construct the ordering "synthetically", i.e., by categorical operations, such as pizero(Aut(T)), applied ultimately to the object T.) I hope this will suggest some clarification of the questions raised by Thomas and Andrej. Bill Lawvere
participants (3)
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Andrej Bauer -
Thomas Streicher -
wlawvere＠buffalo.edu