NB. Freyd characterized the Dedekind reals as a final coalgebra. Alex Simpson and I characterized "the Cauchy completion of the rationals within the Dedekind reals" as a free algebra (to be precise, we started from the algebras as a primitive notion and later found this construction of the free one). But this has already been discussed in postings in the past few years.
Martin Escardo
Does one get any known versions of reals by performing the Cauchy or Dedekind construction starting with initial algebras I for non-decidable lifts L instead of the NNO? It would be then also natural to interpret Cauchy sequences and completeness using appropriate I-indexed families, of course. Even for "integers" Z one has at least three different options: taking I^op+1+I, taking the colimit of I->LI->LLI->..., each map being the unit, or taking the colimit of the corresponding I-indexed diagram. It would be strange if these turn out to be isomorphic. Is any of them an initial algebra for some simple functor? Similarly there are various possibilities for rationals - taking fractions, i.e. a quotient of ZxZ, or colimit of all multiplication maps Z->Z, or of the corresponding Z-indexed diagram. Actually I have not followed the ongoing research for a while, so maybe my questions are outdated. I would be grateful for any references to related work. Mamuka Jibladze