Re: Canonical quotients

[Thorsten Palm <palm@iti.cs.tu-bs.de>]

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/ ]