Re: Canonical quotients

[selinger@mathstat.dal.ca (Peter Selinger)]

[note from moderator: post delayed from Monday by a transmission error]


Hi Bob,

how about this: using a sufficiently strong version of the axiom of
choice, assume a well-ordering on the class of all sets (the "global
well-ordering"). (Someone will be able to tell me whether this is
strictly stronger than the ordinary axiom of choice, or equivalent to
it. In any case, assuming set theory is consistent in the first place,
and sufficiently large cardinals exist, there certainly exist models
of set theory with such a property).

Given any equivalence relation ~ on a set X, say that an element x of
X is a "canonical representative" of ~ if x is the (unique) least
element in its equivalence class under the global well-ordering. Let
X/~ be the set of canonical representatives, and define the map X ->
X/~ to pick out the canonical representative of each class.  I think
this gives canonical quotients on (this version of) Set.

-- Peter

Robert Pare wrote:
>
>
> Peter Freyd's and John Kennison's examples definitively settled
> Mike Barr's question about canonical subobjects that compose. But
> I had started thinking about it and had what I thought would be a
> nice example. The category of sets has canonical quotients (equivalence
> classes) but they don't compose. I think there is no choice that do,
> but so far I haven't been able to prove or disprove this. Anybody?
>
> Bob




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