disjoint_coproducts_?

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

Hello, I have the following question:

Assume a topos SS as the base topos, and  work in this topos as in naive
set theory (without choice or excluded middle). Take a Grothendieck
topos EE ---> SS with a site of definition CC. As usual in the
literature (Joyal-Tierney, Moerdijk, Bunge, and many more) consider that
CC has objects, and that these objects are  objects of EE which are
generators in the sense that given any X in EE,  the family of all
f: C ---> X, all C in CC, is epimorphic. Consider F: CC ---> SS  to be
the inverse image of a point. Then the family Ff: FC ---> FX is
epimorphic in SS.

My question is:

Can I do the following ? (meaning, is it correct the following arguing,
certainly valid if SS is the topos of sets):

Given a in FX, take f:C ---> X and c in FC such that a = Ff(c).

We can break this question in two:

1) Does it make sense to take

E = COPRODUCT_{all f: C ---> X, all C in CC} FC ?

We have g: E ---> FX an epimorphism, so we can take c in E such that
a = g(c).

Then we would need the validity of:

2) Given x in COPRODUCT_{i in I} S_i , then x in S_i for some i in I.

greetings   e.d.


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