What lies behind the definitions of split monics and epics?

Question

Is there an easy way to memorize the definitions of split monics and split epics, and not to confuse the domains/codomains of the arrows from those definitions?

For example, is there a mnemonic rule? And/or what's the motivation for those definitions that would enable one to "deduce" the definition of split monic/epic if one gets lost in domains-codomains involved in the definitions?

I don't understand what lies behind those definitions.

Answer 1

In sets we have two theorems about functions:

Theorem 1. Let $f$ be a function with nonempty domain. The following are equivalent:

1. $f$ is injective.
2. $f$ can be cancelled on the left.
3. $f$ has a left inverse.

Theorem 2. Let $g$ be a function. Assuming the Axiom of Choice, the following are equivalent:

1. $g$ is surjective.
2. $g$ can be cancelled on the right.
3. $g$ has a right inverse.

(The fact that 1 implies 3 is in fact equivalent to the Axiom of Choice; the equivalence of 1 and 2 does not require the Axom of Choice).

When these notions were generalized in category theory, the generalization focused on the second property of each; the reason being that in many instances, those inverses don't exist. For example, in the category $\mathcal{G}roups$ of all groups, one-to-one functions need not have left inverses nor surjective functions right inverses: those are special situations.

However, those special situations are important, as they provide the existence of one-sided inverses. So we still want a categorical way to identify those situations. And those are precisely the "split" cases of monomorphisms and epimorphisms.

Note that if $f$ has a left inverse, then it is certainly left cancellable (hence a monomorphism); and if $g$ has a right inverse, then it necessarily right cancellable, hence an epimorphism. But the converse does not hold.

The "split" cases are the cases that include condition 3 from those two theorems: split monomorphism means "has a left inverse", and split epimorphism means "has a right inverse".

Answer 2

One way to remember is that the maps in a split short exact sequence $$0 \to M \to M \oplus N \to N \to 0$$ are split monic/epic.

In general the identity map $M \oplus N \to M \oplus N$ will not factor through either $M$ or $N$, but the identity maps $N \to N$ and $M \to M$ can and do factor through $M \oplus N$. So, for the monic the factorization $M \to M \oplus N \to M$ tells you that the definition of a split monic is that the identity on the domain factors through the monic.