Recall that a Kan extension is called *pointwise* if it can be computed by the usual (co)limit formula, or equivalently if it is preserved by (co)representable functors.

I have seen pointwise Kan extensions defined in many texts, such as Mac Lane's book or Borceux's book, but these texts don't typically prove any theorems about them. I've also heard it claimed that all mathematically important Kan extensions are pointwise. And apparently it should be easier to compute a pointwise Kan extension since there is a formula for it. But I'd like to know: what are some interesting categorical facts that are true about pointwise Kan extensions that are not true about general Kan extensions? I'm happy with properties whose proofs don't generalize to the non-pointwise case, but I'd also be interested to see actual counterexamples to such generalizations.

In *Basic Concepts of Enriched Category Theory*, Kelly chooses to reserve the term "Kan extension" for the pointwise version. In explaining this choice, he says "Our present choice of nomenclature is based on
our failure to find a single instance where a weak Kan extension plays any mathematical role whatsoever." If I comb through his book for uses of Kan extensions, will I find that "most" facts used about Kan extensions fail in the non-pointwise case?

One fact which I *have* seen stated is that in a Kan extension
$\require{AMScd}$
\begin{CD}
A @>G>> C\\
@VFVV \nearrow \mathrm{Lan}_F G \\
B
\end{CD}

If $G$ is fully faithful and $\mathrm{Lan}_F G$ is pointwise, then the comparison 2-cell is an isomorphism. Is this false when the pointwiseness hypothesis is dropped?

Part of the reason I ask is that there are several definitions in the literature generalizing pointwiseness of Kan extensions to more general categorical frameworks (enriched categories, 2-categories, equipments, double categories...) but I'm confused because I don't know what the theory is that these theories are attempting to generalize. Pointwiseness also shows up in the theory of exact squares, but again I'm confused because I don't know the point of pointwise Kan extensions.

**EDIT**
I'm starting to compile a list of these properties. Here's what I have so far:

As above, a pointwise extension along a fully faithful functor has an isomorphism for a comparison cell.

A functor $F$ is dense if and only if its left Kan extension along itself $\mathrm{Lan}_F F$ exists, is pointwise, and is isomorphic to the identity functor.

To show that taking total derived functors is functorial, one needs that they are pointwise extensions.

To show that the total derived functors of a Quillen adjunction form an adjunction between homotopy categories, one needs to know that they are pointwise extensions.

(3) and (4) were mentioned by Emily Riehl in introductory remarks to a lecture at the Young Topologists' Meeting 2015 (link).

- Pointwise Kan extension along a fully faithful functor is itself a fully faithful functor. More precisely, if $F: A \to B$ is fully faithful, and $\mathrm{Lan}_F G$ and $\mathrm{Lan}_F G'$ exist and and are pointwise, then $\mathrm{Nat}(G,G') \to \mathrm{Nat}(\mathrm{Lan}_F G, \mathrm{Lan}_F G')$ is an isomorphism.

So far the only counterexamples I have to dropping the pointwise hypothesis in these statements are in the cases (1) and (2), which I've detailed in CW responses.

Categorical Homotopy Theory) is the fact that derived functors, when computed via functorial resolutions, are not only pointwise, butabsoluteKan extensions (preserved by any functor whatsoever) -- so they can be computed via the (co)limit formula even though derived categories don't have many (co)limits. $\endgroup$5more comments