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I'm wondering about categorifications of Zorn's lemma along the following lines.

Lemma: if $\mathbf{C}$ is a small category in which every directed diagram of monomorphisms has a cocone of monomorphisms, then there is an object $A$ such that any monomorphism $A \rightarrowtail B$ splits.

Proofsketch: If there were no such object, we could use the axiom of choice to define a function $f$ that assigns to each directed diagram $D$ of monomorphisms, a monomorphism with domain $\mathop{cocone}(D)$ that does not split. Define an ordinal-indexed diagram $D$ by setting $D(\alpha) = f\big( D(\beta) \mid \beta<\alpha \big)$. Because $D$ consists of monomorphisms that don't split, it cannot contain any cycles. But this contradicts smallness of $\mathbf{C}$.

Can this be generalised? (E.g. do we need smallness and monics?) Are other versions known? Are there similar categorical existence statements that are provable without the axiom of choice?

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If the colimits coproiection are monorphisms then your statements follow clearly, do these follow also in general? – Buschi Sergio Nov 19 '12 at 20:12
Good spot. It needs that every directed diagram of monomorphisms has a cocone of monomorphisms. I've edited the question, thanks. – Chris Heunen Nov 19 '12 at 20:21
Tom Leinster has some remarks about Zorn's lemma, wherein he notes that by being very careful with the hypotheses one can prove a (formally weaker) version without choice: – Zhen Lin Nov 19 '12 at 22:19
Aren't you just considering the subcategory of $C$ consisting of monomorphisms, which is then a preorder? A monomorphism that splits in this will be an isomorphism, which is intuitively what Zorn's lemma would give you (a top element, unique up to isomorphism). I don't know if I'd call this categorification, imprecise though that word can be. I would consider taking a (2,3)-category and trying to formulate a Zorn-style result as categorification (what you have, with my spin on it, is a (1,2)-category). – David Roberts Nov 19 '12 at 22:41
David, left-cancellative categories (i.e. categories in which every morphism is monic) are not necessarily preorders; think of any groupoid. Having said that, it would definitely be interesting to "derive" a lemma such as that in the question from Zorn's actual lemma applied on a "higher" categorical level; could you make that more precise? – Chris Heunen Nov 19 '12 at 23:04

In their book Categories and Sheaves, M. Kashiwara and P. Schapira prove the following theorem (Theorem 9.4.2) :

Let $C$ be an essentially small non empty category which admits small filtrant inductive limits. Then $C$ has a quasi-terminal object, that is, an object $X$ such that every morphism with source $X$ has a left inverse.

It seems to me that the hypothesis is stronger than yours, but the conclusion is also stronger. (What about considering the subcategory $C'$ of $C$ whose arrows are monomorphisms of $C$?)

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