in which conditions the following holds: If the pullback of a morphism is an isomorphism, then this morphism is an isomorphism?

Question

It is easy to see that, if p is a split epimorphism, the the following proposition is true:

"If the pullback of q along p is an isomorphism, then q is an isomorphism".

Is there an weaker condition on p to which the proposition above would also be true?

Answer 1

It is almost equivalent to the condition that p is an extremal epimorphism. A few months ago I made some notes about this condition:

http://arxiv.org/abs/1311.2974

(related MO question: https://mathoverflow.net/questions/143070/two-pullback-diagram)