Kan extension are an incredibly useful concept when they exist. My question is: can we still derive information about a functor when Kan extensions don't exist, and if so, what information and in what way?


I can think of one way to look at it:


Let $p:C\to C'$ and $F:C\to D$ be functors. We can define a category of pairs $(G:C'\to D, \eta: F\Rightarrow p^\*G)$ with morphisms being natural transformations that commute with the $\eta$s.


A left Kan extension of $F$ along $p$ is an initial object in this category, so one can study this category as a replacement when the Kan extension fail to exist.


Has this direction been explored before?



When Kan extensions don't exist
Yes, directions like this have been explored, for all kinds of objects with universal properties (which includes Kan extensions, since as MacLane famously wrote "all concepts are Kan extensions"). Rather than the category whose initial object would be the missing object, it's more common to consider the functor for which a representation would be the missing object. Properties of this functor can then be considered as properties of the "missing object", and at the extreme the category of all functors/presheaves (or a subcategory such as a category of sheaves, concrete sheaves, schemes, etc.) can be studied as a "better-behaved" replacement for the original category in which "missing objects" have been put in.


As one concrete example, a non-existing left adjoint can exist as a [pro-adjoint](https://ncatlab.org/nlab/show/pro-left+adjoint); in fact it does so precisely when the putative right adjoint preserves *finite* limits (whereas, modulo size considerations, for an actual left adjoint to exist, the putative right adjoint would have to preserve *all* limits).


