Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early introduction in 1960, and taking into account the extensive use of category theory by Grothendieck and his students, I am curious whether this mathematical school employed Kan extensions in their work and where.
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1$\begingroup$ They're plentiful in SGA 4, but just not called by that name. The inverse image functor on sheaves is a left Kan extension (followed by sheafification). At any rate, since it's just some explicit limit or colimit construction, I don't really see what they would gain from using this name. $\endgroup$– R. van Dobben de BruynCommented Nov 28, 2023 at 21:08
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$\begingroup$ The final section of Categories for the Working Mathematician is called All concepts are Kan extensions. Well, yes and no. No, because there are many essentially equivalent concepts. including adjunctions, universal properties, but also the formation, introduction, elimination, $\beta$ and $\eta$ rules of a type theory. Which of these you use for a particular problem is a matter of context and also personal taste. But, if you regard all of these ideas as avatars of the same concept, then yes, what Saunders Mac Lane said is correct. $\endgroup$– Paul TaylorCommented Dec 10, 2023 at 17:28
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