Zorn's lemma in categorical language

Question

The axiom of choice, which is equivalent to Zorn's Lemmma, has a nice categorical "translation": in the category of sets, every epi is a retraction.

So the axiom of choice says something about the structure of the category of sets.

Can Zorn's lemma be equally simply translated into categorical language? For example, does Zorn's lemma say something about the category of posets (and order preserving functions) or another well-known category.

Answer 1

This might not be exactly what you are looking for, but, since any poset can be made into a category by making the elements of the poset the objects of the category and making exactly one morphism $a\rightarrow b$ if $a\le b$, and none otherwise, we can turn Zorn's lemma into a statement about categories.

Here is a translation of Zorn's lemma into categorical language, using this correspondence:

If $\mathcal{C}$ is a poset and, for any totally ordered set $\mathcal{I}$ and functor $F: \mathcal{I} \mapsto \mathcal{C}$, $F$ has a colimit cocone, then there is an object in $\mathcal{C}$ whose only outward morphisms are isomorphisms.

You can't just use the more standard categorical notion of final object, because Zorn's lemma only guarantees the existence of a maximal object, not a maximum one. (Thanks to Zhen Lin for pointing this out).