Re: Reply to Eduardo Dubuc

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

Dear Eduardo,

I understand your point better now. For you FX is already given as the
respective colimit. This was shown by Celeyrette in his 1974 Th'ese
directed by B'enabou (Ch.VI Theorem de Kan where you also find a proof
of Diaconescu's theorem over arbitrary bases).
But you have to rely on a theorem characterizing colimits of small
diagrams in SS from which it follows that the source tupling (in the
sense of internal sums) of the cone is epic.
Even if you think it is splitting hairs you cannot express the
property of being epic in SS within the internal language of SS. You
can express in the internal language of SS that a map is surjective
and show externally that SS validates f being surjective iff f is epic
in SS. (For expressing "epic" in SS one would have to quantify over
all objects in SS which one cannot do in the internal language of SS.)

So my conclusion is that the argument you give is ok for most purposes.
What I wrote above (and previously) gives some additional justification
of its correctness.

I will be offline till next weekend and therefore have no time to
continue a discussion with many mails back and forth.
Moreover, I think things have been clarified to quite some extent.

Best, Thomas

PS "purely external categorical construction in SS"
is a contradiction in terms
formulations like these have given me the impression that you have
wrong reason for a correct result



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