From: Peter Freyd <pjf@saul.cis.upenn.edu>
It could well be that Vaughan and I are defining the midpoint structure in the same way.
Yes, after changing g(dx,uy) to g(ux,dy) in line 2 of Peter's definition of g and similarly for line 3 (otherwise g(dx,uy) simplifies to the nonsensical g(T,B)) Peter and I have essentially the same coalgebra on IxI, and exactly the same after some inessential permutations within that definition. While I rather like Peter's nonrecursive definition T v F v B F'v F' F' I --> 1 v I v 1 ------> I v I v I v I ---> I v I --> I of the halving map h:I --> I sending x to x/2, it should be remarked that the effect of this map as an operation on sequences is to preserve the empty sequence, and for nonempty sequences simply to prepend a copy of the leading digit, e.g. -++-+000... becomes --++-+000.... (This takes the 3-symbol alphabet for Peter's number system to be {-,0,+}.) In other words, right shift by one with sign extension, a well-known realization of halving. Along the same lines, Peter's d and u maps shift the sequence left. If d (resp. u) shifts a + (resp. -) off the left end, the result is replaced by the constantly + (resp. -) sequence, i.e. "clamp overflow to +1 (resp. -1)". Although the interval [-1,1] goes naturally with Peter's final coalgebra, it occurrs to me that a fragment of Conway's surreal numbers is perfectly matched to it, namely the interval [-\omega,\omega] consisting of the real line plus two endpoints. With respect to the Conway story this can be described as what Conway produces by day omega, modulo infinitesimals (identify those surreals that are only infinitesimally far apart). At no day does exactly the real line appear in Conway's scenario. Prior to day omega only the finite binary rationals have appeared. Day omega sees the sudden emergence of all the reals along with 1/omega added to and subtracted from each rational, as well as -omega and omega. Except for -omega and omega, the quotienting eliminates the 1/omega perturbations, yielding exactly the real line plus endpoints. Exactly the same quotienting happens with Peter's final coalgebra, whose elements are representable as finite and infinite strings over {-,+} (the 0 is eliminated by allowing strings to be finite; if you want all strings to be infinite, put 0 back in the alphabet and use it to pad the infinite strings to infinity). For example ---+++++... and --+----... are identified by both Conway and Freyd. Using Peter's choice of [-1,1] as the represented interval, these are both -1/4. Using Conway's number system, these are respectively -2 - 1/omega and -2 + 1/omega, which with the identification I described above become -2. In Conway's setup the finite constant sequences are the integers, with the empty sequence being 0 and counting being done in unary. At the first sign reversal the bits jump mysteriously from unary to binary, not by fiat but as a surprising consequence of a definition of addition that on the face of is so natural that you would not dream it could cause a radix jump like that. So both [-1,1] and [-omega,omega] each admit a natural final coalgebra structure for Peter's functor, namely Peter's and Conway's respectively, and I would be surprised to see a different final coalgebra in either case that was as natural. In contrast, Dusko and I exhibited a number of more or less equally convincing final coalgebra structures on [0,1) and [0,omega) for the functor product-with-omega, no one of which I would be willing to call *the* right one. Vaughan