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  • $\begingroup$ After posting this, I started searching around and found Kan injectivity in order-enriched categories by Adámek–Sousa–Velebil, where they study objects in order-enriched categories that are "Kan-injective" with respect to a class of morphisms, and show them to be characterised as cocomplete objects. Presumably their development could be carried out at the same level of generality as in this answer to give a more general understanding of Kan-injectivity. $\endgroup$ Commented May 1, 2021 at 13:56
  • $\begingroup$ Thanks, this is interesting! I hadn't thought to ask for the extension $\mathcal B \to \mathcal K$ to have a universal property like being a Kan extension. So I suppose this "Kan-injectivity" is stronger than "lax-injectivity" and a priori incomparable to pseudo-injectivity. Although, I'm thinking that because $I \to I^\triangleright$ is fully faithful, and because admitting LKE along such functors is equivalent to being cocomplete, doesn't the conclusion "left-Kan-injective iff cocomplete" follow pretty directly, without invoking any general KZ theory? $\endgroup$ Commented May 1, 2021 at 14:43
  • $\begingroup$ @TimCampion: ah yes, I took the liberty of choosing a stronger definition of extension than in your question – it seemed like the natural generalisation from the posetal case considering the relationship between LKEs and cocompleteness! I'm not sure I knew of this property regarding left extensions along $I \to I^\rhd$, which does greatly simplify this answer – but I think in any case it's more satisfying to have a more conceptual understanding that generalises more easily :) $\endgroup$ Commented May 1, 2021 at 15:25
  • $\begingroup$ Definitely, I like this setting. But in the poset case, what I find surprising is that the injectivity characterization of complete posets does not require that the extension have any universal property (though I suppose a "Kan-injectivity" characterization also holds in that case). One proof: if $K$ is a poset which is strictly injective with respect to embeddings, then $K$ is a retract of its Dedekind-MacNeille completion. Then apply the Knaster-Tarski theorem: if $f: P \to P$ is any order-preserving endomorphism on a complete poset $P$, then $Fix(f)$ is a complete poset. $\endgroup$ Commented May 1, 2021 at 15:30
  • $\begingroup$ @TimCampion: that's true. I had been thinking that there was a strong connection between the injectivity characterisation and the Kan injectivity characterisation, which would justify interchanging them, but now that I think about it, it's not so clear. It would be good to have something like Proposition 4.4 of Escardó's Injective spaces via the filter monad, which could justify Kan-injectivity in this setting. $\endgroup$ Commented May 1, 2021 at 15:47