When every epi splits a category is said to satisfy the Axiom of Choice.

When every idempotent splits a category is called Cauchy Complete or Idempotent complete.

These look to be well-studied notions, but what about the corresponding question for monos, i.e., the Axiom of Choice for the opposite category? Does it have a name and are there any references on it? Is it trivial in some way I'm not seeing and that's why it's not studied?

As examples, it's true in Vect, and clearly it's true for categores of NonEmpty and Pointed Sets (using at least LEM).

Axiom of Choiceto atoposin which all epis split because a topos has other structure capable of making this agree with other forms of AC. However, Max doesn't say anything else about the structure of his category or give any hint of applications besides that monos split. $\endgroup$isa big difference in my setting (Programming Languages) where these will almost always be distinct notions. So unfortunately I have no applications for this idea, only non-applications. $\endgroup$1more comment