Kan extensions and orthogonality

Question

$\require{AMScd}$I have the feeling that there is a connection between the existence of a Kan extension in a 2-category ($\bf Cat$ will be sufficient) and a 2-dimensional notion of orthogonality.

Given $$\begin{CD} A @>f>> B \\ @VgVV @.\\ C \end{CD}$$ the existence of $h = lan_gf : C \to B$ still "solves the lifting problem" or "shows that $B$ if $g$-local" (according to your preferred terminology), but only in a weak sense, because $\eta : f \to hg$ is in general neither an identity, nor invertible (although it is still universal).

I am reinforced in this feeling by a few examples:

1. here https://arxiv.org/abs/1503.06469 lax orthogonal factorization systems are defined, and although the 2-dimensional aspect is kept to a bare minimum (hom-categories are simply posets), lax orthogonality is defined exactly via an extension property (see first lines of page 2);
2. In this paper "(ptwise) cocomplete 0-cells" $X$ are defined as those objects of a 2-category $\cal K$ that admit all (ptwise) left extensions; the class of all these $X$'s is treated, there, exactly the same way as an orthogonal class (with the aim to prove, under surprisingly mild assumptions, that it is 2-reflective, i.e. that "cocompletions exist").
3. In the diagram $$\begin{CD} A @>f>> B \\ @VgVV @.\\ C \end{CD}$$ of morphisms in a 1-category, orthogonality is attained precisely if $g^*=\hom(g,B)$ is invertible. Here, the (say, left) Kan extension of $f$ along $g$ exists if $g^*$ has a left adjoint.

So,

How do I make precise this intuition? How do I ground it on these examples? Does the notion of weak orthogonality, of which "being $g$-local" is a particular case, coincide with the lax orthogonality defined in the first paper (in such a way that, say $lan_g$ exists when $B\to *$ is right lax orthogonal to $g$)?

Answer 1

My impression is that this is essentially the topic of Di Liberti–Lobbia–Sousa's KZ-pseudomonads and Kan Injectivity. Connections to lax orthogonal factorisation systems have been explored in unpublished work by Bourke and Walker (see Walker's slides Characterizations of lax orthogonal factorization systems).