Reals and rationals

Reals and rationals

[Jiri Adamek]

This is a comment to the recently discussed Pavlovic-Pratt paper: M. Barr
has proved in "Terminal Coalgebras ..."( TCS 114) that for bicontinuous
set functors initial algebras carry a natural metric, and a Cauchy
completion of that space is a final coalgebra. This result can be
generalized to endofunctors F of any locally finitely presentable
category. Side conditions: F preserve monomorphisms and have a point in
F(O). Of the two functors used by Dusko and Vaughan, only F_2 satisfies
the latter, of course. Its initial algebra is the set of all rationals in
[O,1), which illustrates the idea quite well. The statement of the
generalization I have in mind is that the structure of a final coalgebra T
(which in a locally finitely presentable category is determined by
morphisms from all finitely presentable objects B into T) is determined by
the structure of an initial algebra I in the following sense: each
hom(B,I) carries a natural metric and a Cauchy completion is hom(B,T).
Thus, for endofunctors of Pos not only the elements of T but also its
order is obtained by completing I .

Jiri Adamek