One little point: in my first 1999 posting on the subject I wrote: In fact, we didn't need to start in the category of posets. It would have sufficed to work in the category of sets with distinct top and bottom...The final coalgebra is still the closed interval and, yes, the ordering is implicit. But the first explicit final co-algebra characterization of the closed interval as a _space_ seems to have came a month later: Date: Mon, 24 Jan 2000 20:14:36 +0100 (MET) From: "Martin H. Escardo" <Martin.H.Escardo@ens.fr> To: categories@mta.ca Subject: categories: Freyd's couniversal characterization of [0,1] It would be interesting to test Freyd's couniversal characterization of the unit interval in many other categories. Here I test it in Top, the category of topological spaces and continuous maps, and various full subcategories, where one would hope to get the unit interval with the Euclidean topology. ---------------------------------------------------------------------- Summary of the outcome of some tests: (1) In Top, the final coalgebra for Freyd's functor exists. Its underlying object, however, is an indiscrete space (unsurprisingly). (2) In the category of T0 spaces, it doesn't exist. (3) In the category of normal spaces it does exist, and, as one would hope, its underlying object is indeed the unit interval with the Euclidean topology. ---SNIP--- For the rest see north.ecc.edu/alsani/ct99-00(8-12)/msg00082.html There was an important follow-up: Date: Thu, 27 Jan 2000 13:25:25 +0100 (MET) From: "Martin H. Escardo" <Martin.H.Escardo@ens.fr> To: categories@mta.ca Subject: categories: re: Freyd's couniversal characterization of [0,1] The following is of course rather unpleasent:

(1) In Top, the final coalgebra for Freyd's functor exists. Its underlying object, however, is an indiscrete space (unsurprisingly).

This can be fixed by choosing a slightly different category of bipointed objects. Define a *regularly bipointed object* to be an object X with two distinguished points x0,x1:1->X such that [x0,x1]:1+1->X is regular mono. Then the terminal coalgebra for Freyd's functor is 1 iff 1+1=1. With the restriction to regularly bipointed topological spaces, the statement (1) becomes false because the two-point discrete space 1+1 is not homeomorphically embeded as a subspace of any indiscrete space. Hopefully, there isn't a final coalgebra in RegBi(Top), but I don't see any if there is, it cannot be the Euclidean interval. ---SNIP--- 26-Nov-2004 14:25:47 -0400,3203;000000000000-00000000