Subject: categories: Re:arithmetical and geometric reals in (models of) SDG

From: [email-masked]
Date: Sat, 10 Apr 2004 18:28:20 -0400

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