hi. i somehow managed to unsubscribe from this list at some point, and almost missed the thread about coalgebra of reals. i am not sure that i have seen all the relevant postings, but here are my 2p of thoughts. if i am not misunderstanding anything, peter freyd's closed interval coalgebra achieves something that vaughan pratt's and mine do not. we work with lists and streams and get *irredundant* representations of reals. however, without redundancy, the algebraic operations on reals are undecidable. peter's bipointed approach, however, induces a *redundant* representation. this allows him to extract the algebraic operations coinductively. in the talk at the 1998 lisbon workshop on coalgebra (and later in some seminar talks), i described a redundant coalgebra of conway reals, where conway's definitions of the field operations were derivable as coalgebra homomoprhisms. however, the setting seemed too complicated, probably due to me. peter's definition of the midpoint algebra is considerably simpler, although the resulting operations are probably still computationally quite inefficient. the coalgebra of alternating dyadics, described in the paper with vaughan, was derived from this 'conway' coalgebra. as explained in sec 3.4, it provides the best dyadic approximation, while the continued fraction coalgebra, provides the best rational approximation. presumably, it can be derived from the 'norton' coalgebra (contorted fractions), mentioned by peter johnstone. there are as many redundant coalgebras of reals, as there are corresponding irredundant ones (one for every interval partition), for fast outputs. i think that some of the mentioned *mysteries* about the maps between these various representations boil down to the fact that they are coalgebra isomorphisms, that do not preserve the derived structure, but rather *transform* it. eg, the laplace transform is a coalgebra isomorphism between the coalgebra of analytic functions and a dual coalgebra of meromorphic functions --- whereby the convolution ring gets transformed into a multiplicative domain, etcetc. in any case, when you are done with reals, you may be interested to have a look int my paper with martin escardo "calculus in coinductive form", LICS98, or ftp://ftp.kestrel.edu/pub/papers/pavlovic/LAPL.ps.gz all the best, -- dusko PS along the lines of peter's bipointed construction (and also to check whether i understood it): * bipointed sets are the algebras for the functor 2+(-) : Set -> Set. * bipointed sets with two _distinct_ points are the free such algebras. so lets work in the kleisli category *K* for 2+(-). write 2 = {0,1} * the bifunctor (-) v (-) : *K* --> *K* maps * the objects X, Y to X + {m} + Y * the arrows X --f-->2+U and Y--g-->2+V to h : X+{m}+Y ---> 2 + U+{m}+V h(x) = f(x) if f(x) = 0 or f(x) in U h(x) = m if f(x) = 1 h(M) = m h(y) = g(y) if g(y) = 1 or g(y) in V h(y) = m if g(y) = 0 * now (0,1) is the final coalgebra for the functor X v X on *K*. the coalgebra structure (0,1) ---> 2+(0,1) +{m}+(0,1) maps 1/2 to m, x<1/2 to 2x and x>1/2 to 2x-1. * the finality is now easy to prove directly, by displaying the anamorphism for any given coalgebra X -r-> 2+X+{m}+X. ditto for the algebraic structure.