Re: Canonical quotients

[Michael Barr <barr@math.mcgill.ca>]

But that's only an equivalent category; everyone knows that can be done.
Even easier is the dual category of complete atomic boolean algebras,
which has canonical subobjects.

On Sun, 12 Sep 2010, Thorsten Palm wrote:

>
> Robert Pare hat am 12.09.10 geschrieben:
>
>>
>> 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?
>
> There is. First consider the full subcategory of partitions; that is,
> sets whose elements happen to be non-empty, pairwise disjoint sets. It
> has an obvious choice of quotient maps that does the trick, namely
> those maps for which each element of the target is the union of its
> fibre. For the remaining sets as sources, additionally choose the
> identity in case of the trivial quotient, the canonical map otherwise.
>
> Thorsten
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]