categories: Re: Real interval halving

[Vaughan Pratt <pratt@cs.stanford.edu>]

It's true I only learned about the Artin(?)-Schanuel construction
recently---Peter Freyd mentioned it to Phil Scott and me at lunch one
day at LL'96 in Tokyo, albeit with yet another attribution, Conway.
I thought it was very cute.

There is some sort of duality that I don't understand between this
construction and the Knuth-Pratt construction.  The former puts
the bounded sequences in the denominator and does the work in the
numerator, namely requiring that s(m+n) = s(m) + s(n) to within a
constant independent of m and n.  The latter puts the bounded sequences
in the numerator and does the work in the denominator, namely modding
out by the equation x/2 + x/2 = x, where x/2 is defined as right shift
(prepend zero).

Artin-Schanuel can be related to Dedekind as follows.  For Dedekind,
the unreduced rationals m/n together with all m/0 constitute Z^2,
the lattice points of the plane, whose nonempty rays are then the
reduced rationals.  The irrationals along with infinity (thinking of
the real line projectively) are then obtained as the empty rays, all
of which make distinct Dedekind cuts in the rationals, qua rays *or*
qua irreduced rationals (actually two cuts are needed in the projective
line, infinity supplies the other).  Artin-Schanuel identifies all but
the ray at infinity in terms of their neighborhoods instead of cuts,
specifying one point of the neighborhood per column.

Moving the bounded sequences from the denominator to the numerator
doesn't seem to be compatible with this picture.  What picture if any
is it compatible with?  And is there a categorical duality that goes
along with this inversion?  A passage from algebra to coalgebra perhaps?
Or is it the duality of addition and multiplication---we have
s(m+n) = s(m) + s(n) doing the work above and x/2 + x/2 = x at work
underneath, albeit with addition also party to the latter.

Vaughan