Balanced

Suppose that in a Category we have: Monomorphism Epimorphism Isomorphism Then we say that is balanced.

Lemma

If is balanced, then any Separating Family is also a Detecting Family.

Proof

Let be a Separating Family. Suppose is such that every with factorizes uniquely through . Suppose

for some . Let with Then

so

because factorizes uniquely through . This holds for any with and as is a Separating Family, we conclude

and thus is a Monomorphism.

Now suppose

Any has a unique such that so we can multiply to find

for any with and thus so is an Epimorphism. As is balanced, then is an Isomorphism and so is a Detecting Family.