Back here John wrote

But this doesn’t really explain why people talk about Kan extensions a lot and Kan lifts very little.

My rough impression is that Kan extensions show up very often in contexts where one has a rich environment $D$ (I’m following Jim Dolan in his video lectures on algebraic geometry, using the word ‘environment’ to refer to codomains: receiving categories in which ‘theories’, or domain categories, are interpreted), and one wants to extend one ‘model’ $m: C \to D$ of a ‘theory’ $C$ to a model of another theory $C'$ along a theory morphism $t: C \to C'$, and do this in a universal way.

An archetypal example of this is left Kan extension along a Yoneda embedding: given a functor $f: C \to D$ (with $C$ small), the left Kan extension

$\hat{f} = Lan_{y C} f: Set^{C^{op}} \to D$

exists provided that $D$ is cocomplete. In other words, an interpretation $f$ of a theory $C$ in an environment $D$ can be extended (universally) to an interpretation $\hat{f}$ of a richer theory $\hat{C} = Set^{C^{op}}$, if the environment $D$ supports that. This basic example extends in a variety of directions; for example the theory $C$ could be a monoidal or symmetric monoidal theory, and the interpretations are to preserve the monoidal structure, and we consider the extension to a richer monoidal or symmetric monoidal category $Set^{C^{op}}$, obtained by freely adjoining colimits (= weights on $C$).

In any case, Kan extensions are from this point of view about extensions of models, extending a model of one theory to a model of another, but within the same environment.

What about Kan lifts? This time the theory $C$ remains the same, but we change the environment, i.e., we have a situation like

$\array{
& & D' \\
& & \downarrow p \\
C & \overset{f}{\to} & D
}$

where we are trying to lift an interpretation $f$ of $C$ in $D$ to a (maybe richer) environment $D'$ – for example, $D'$ may be a category of algebras over $D$. Now the Kan lift may very well exist and be important, but maybe we recognize it by another name. For example, suppose we want the left Kan lift of a functor $f$ through $U$ in the situation

$\array{
& & M-Alg \\
& & \downarrow U \\
C & \overset{f}{\to} & Set
}$

where $M$-$Alg$ is the category of algebras of a monad $M$, and $U$ is the underlying set functor. Then the left Kan lift exists, but it’s obtained just by composing $f$ with the free functor $F: Set \to M-Alg$. In other word, the lifting problem is solved in a trivial way, so we don’t particularly refer to it as a Kan lift – it just is what it is.

So to wrap up this story: part of the asymmetry here may be due to the fact that in practice, theories are often “small” (e.g., sites), whereas the environments in which they are interpreted are often “big”, or “rich” – categories like $Set$ or perhaps a sup-lattice like $\mathbf{2}$. And whereas Kan extensions draw attention to themselves by being constructed by a process which is interesting in its own right, Kan lifts don’t stand out so much because they are often effected by some simple process like postcomposing with some arrow which is “there” because the environments accommodate it.

I’m obviously not attempting a complete explanation here, and obviously some of this is to be taken with a grain of salt.

## Re: Kan Lifts

I haven’t been aware of “Kan lift” as a technical term.

Am I right in guessing that what is meant is just Kan extension in an opposite 2-category?

Googling, by the way, reveals experts on Kan lifts here