Kan extensions in concrete 2-categories

Question

Kan extensions make sense in any 2-category. I am interested in Kan extensions in "concrete" 2-categories consisting of actual categories with some sort of structure (e.g., finite products, finite limits, pretoposes, toposes, etc.), with 1- and 2-cells structure-preserving functors and any natural transformations.

First of all, when do such extensions exist? The usual Kan extensions exist whenever the target category is co-complete, but these functors do not typically preserve the relevant structure (aside: what are some relatively simple examples of this phenomenon). Is there some way to "fix up" this problem and provide a "structure-preserving Kan" given co-completeness? If not, are there other conditions which might allow us to do that? Failing either of these, are there at least conditions under which the ordinary Kan extensions will preserve structure X?

This leads to a second question. If (some) extensions do exist in a concrete 2-category, how are they related to the usual extensions in $\textbf{Cat}$? Intuitively, I would expect the forgetful functor to preserve right Kan extensions, as these are akin to limits, and the free completion to preserve left Kan extensions.

Thanks!

Answer 1

One situation in which a Kan extension can be "fixed up" is if the category of structure-preserving maps between two structured categories is a reflective or coreflective (full) subcategory of the category of all maps. For instance, if $C$ is small and $D$ is well-behaved (e.g. locally presentable) and both have finite limits, then $\mathrm{Lex}(C,D)$ is a reflective subcategory of $[C,D]$. Thus, applying the reflector to a left Kan extension in $\mathrm{Cat}$ will "fix it up" into a left Kan extension in the 2-category $\mathrm{Lex}$.

I spent a few minutes thinking about whether anything general can be said about for what sorts of "structure" this is the case. A natural question to ask is whether it is always true when our structured categories are the algebras for a lax-idempotent or colax-idempotent 2-monad. However, I don't have an answer.

Answer 2

These are the sorts of results that should be in Verity's thesis. The general idea is: suppose we have two "category theories" with some form of 2-functor between them; then what categorical structures like (co)limits and Kan extensions are preserved by the 2-functor?

To formalize the notion of a "category theory", Verity uses a version of proarrow equipments. I believe that Verity defines a suitable notion of adjunction of 2-functors between proarrow equipments (this is really a 3-categorical concept) which he refers to as a biadjunction. And I believe that Theorem 1.7.1 in the above-linked thesis has the sort of result you want: certain colimits (including: certain left Kan extensions) are preserved by right biadjoints. I'm not certain whether the correct dual statement is that left biadjoints preserve left Kan extensions, or right Kan lifts. I'm also not clear enough on biadjunctions between equipments, but I expect that the obvious adjunction between categories and finitely complete categories is an example.

Mike Shulman can probably do this with a stricter version of proarrow equipments and biadjunctions between them. Hopefully he or another expert will have something more informed to say.