# What lies behind the definitions of split monics and epics?

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.

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".

• This wasn't part of the my question, but your answer made me wonder if there is a similar motivation for regular monics/epics. Monics/epics correspond to the second points in your theorems; split monics/epics correspond to the third. Do regular monics/epics correspond to the first bullet points in some way? – user634426 Feb 17 at 5:46
• @user634426: The first point is what the concept corresponds to in the category of sets. For concrete categories, if the underlying function satisfies 1 then it satisfies 2; so an injective function will necessarily be a monomorphism, and a surjective function will necessarily be an epimorphism. But the converse does not hold even for concrete categories. The first bullet point is the motivation. – Arturo Magidin Feb 17 at 6:38
• I see, so "regular monic" does not quite fit into the context of Theorem 1 or 2. But is there anything else that would motivate (or that would be parallel to) the notion of regular monic and regular epic (just like the second and third bullet points of Theorem 1 are parallel to the notions of monic and split monic, respectively)? Why is the notion of regular monic/epic important (I assume it is, since it has a separate name). – user634426 Feb 17 at 6:56
• @user634426 One could simply add a fourth condition: $f$ is an equalizer, respectively a coequalizer. This is perhaps less satisfying, but it’s harder to even talk about regular monos and epis without introducing universal properties. The motivation comes more from abstracting the properties of kernels to normal monos, then regular monos, than from extracting a known property of injective functions. Perhaps from the difference between inclusions of subspaces and arbitrary injective continuous functions, too. – Kevin Carlson Feb 17 at 8:46
• @user634426: I misunderstood your reference to "regular monic" as being "the usual concept of monic". Sorry about that. Yes; you would need to add a fourth equivalence, but as far as I know the notions of equalizer and coequalizer are later notions. – Arturo Magidin Feb 17 at 15:33

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.