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Most Kan extensions arising in nature are pointwise, and this observation prompts Kelly to write [1]:

Our present choice of nomenclature is based on our failure to find a single instance where a [nonpointwise] Kan extension plays any mathematical role whatsoever.

I am interested in counterexamples to this (implicit) claim: namely, examples of Kan extensions that play some mathematical role but are not pointwise. This is, of course, a somewhat loosely formulated question, in that "playing some mathematical role" isn't a well-defined property, and it is straightforward to come up with examples of nonpointwise Kan extensions, whose mathematical significance may be attempted to be justified a posteriori. To restrict the scope a little, I am primarily interested in nonpointwise Kan extensions arising from motivations in pure category theory, i.e. that play a role in an abstract proof or construction, rather than providing a specific construction in a specific category.

To put it another way: are there examples of pointwise Kan extensions that might have convinced Kelly that they were worthy of consideration?

[1]: Basic Concepts of Enriched Category Theory

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    $\begingroup$ I don't know many non-pointwise Kan extensions, and ones I do know don't seem to play any mathematical role. $\endgroup$
    – Tim Campion
    Commented Feb 10, 2022 at 14:12
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    $\begingroup$ Actually -- you say that it's straightforward to come up with examples of non-pointwise Kan extensions in general (when you don't ask for them to be "mathematically significant"). I find this claim interesting -- how do you do this? And what kinds of examples do you end up with? $\endgroup$
    – Tim Campion
    Commented Feb 10, 2022 at 14:22
  • $\begingroup$ Do you count derived functors, which are classically defined to be not necessarily pointwise but in practice not merely pointwise but even absolute? $\endgroup$
    – Zhen Lin
    Commented Feb 10, 2022 at 14:27
  • $\begingroup$ @TimCampion: I meant that we can construct Kan extensions specifically for the purpose of being examples of nonpointwise Kan extensions (one example being Exercise 3.9.7 of Borceux, though your examples probably also qualify in this sense), rather than being constructed for some useful purpose. $\endgroup$
    – varkor
    Commented Feb 10, 2022 at 18:16
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    $\begingroup$ I'm not familiar with derived functors used in practice. In general theory – model categories, derived categories of chain complexes with K-injective resolutions, deformable functors on homotopical categories, etc. – it seems to me that when derived functors exist for general reasons they are absolute Kan extensions. Maltsiniotis advocates taking absoluteness as part of the definition in order to get adjointness for free. $\endgroup$
    – Zhen Lin
    Commented Feb 10, 2022 at 22:50

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