Zorn's lemma in categorical language

Solution 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).