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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.

My previous proof of (2) doesn't work with the proposed modification,
but this should still hold. The argument for (3) is undisturbed.

 >     (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.

Martin Escardo