This is a follow up to my post from yesterday.
Let me say that a subobject R in A x B is a Mal'cev relation if
R o R\op o R is in R. Call a Mal'cev relation effective if there is a
pullback
R ------> A
| |
| |
| |
v v
B ------> C
Yesterday, I pointed out, in effect, that in toposes and abelian
categories, Mal'cev relations are effective and asked if it was true in
the dual of sets. The following additional facts have now come to
light:
1. Mal'cev relations are not effective in the dual of sets.
(Counter-example at end).
2. In a Mal'cev category, every relation is Mal'cev (trivial), but
not always effective (see 1).
3. If every Mal'cev relation is effective, then so is every equivalence
relation. Since equivalence relations are effective in the opposite of
sets, it follows from 1 that the converse is false. Thus effectiveness
of Mal'cev relations is strictly stronger than that of ERs.
4. A PER (on a single object A) is a Mal'cev relation, but the converse
is not true (trivial counter-example in groups, in which, from 2, every
relation is Mal'cev).
Thus the condition of effectiveness of Mal'cev relations is an exactness
condition that is strictly stronger than that of ERs. Since it is true
in any topos, it must have something to do with disjoint universal sums.
It wouldn't surprise me if it had something to do with effective unions.
Here is the counter-example. If there is a pullback as above, then C
can be taken to be the pushout. Thus if sets\op satisfy the condition,
so does the opposite of finite sets, which is equivalent to finite
boolean algebras. In that category, the following is a pushout and not
a pullback (there is no choice in the maps since 2 is initial and 1 is
final):
2 ------> 4
| |
| |
| |
v v
1 ------> 1
Since Boolean algebras is a Mal'cev category, all relations are Mal'cev
(it is trivial to see that this one is in any case).
It follows that effective Mal'cev pair would be another answer to the
original question, but not a good one, I think.
Michael
==========================================================================
