Sufficient conditions for all regular epimorphisms in a category to split

Question

The axiom of choice internal to a category is the statement that all epimorphisms in that category are also split epimorphisms. I am interested in sufficient conditions under which the weaker statement "all regular epimorphisms split" is true. Obviously the converse holds in every category.

Optimally I would like conditions that are necessary and sufficient, but conditions that are stronger than "all regular epimorphisms split" but weaker than "all epimorphisms split" are also welcome.

Answer 1

I don't quite know what you have in mind, but here goes something (this is mostly but not completely standard):

Let $\mathcal E$ be a set (/class) of epis, in your case let it be the set of all regular epis.

You can also consider strong, extremal and, in pointed categories, normal epis (and possibly others). If you are keen on that you can also define "normal epis relative to a coequivalence corelation" (corelation, not correlation) in a non-pointed category, but I have no idea if this has any uses at all. Obviously the larger class, the stronger the choice-property.

Anyway:

An object $P$ is $\mathcal E$-projective if for all $e : A \to E \in \mathcal E$ and $p : P \to E$ there exists a morphism $x : P \to A$ such that $e\circ x = p$.

An object $P$ is $\mathcal E$-choice if every $e : A \to P \in \mathcal E$ is split epi.

Equivalent are:

1. the "axiom of $\mathcal{E}$-choice", i.e. every $e\in \mathcal{E}$ is split epi
2. every object is $\mathcal{E}$-projective
3. every object is $\mathcal{E}$-choice

Proof: ... is easy, briefly:

1. $\Rightarrow$ 2. : Take $x := m\circ p$ where $m$ is a section of $e$.
2. $\Rightarrow$ 3. : Take $p = \operatorname{id}_P$. The morphism $x$ is then a section of $e$.
3. $\Rightarrow$ 1.: Immediate.