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I trying to decide whether the existence of a, say, a left Kan extension along $i: C \to D$ for any functor $F: C \to E$ implies that $i$ has a right adjoint.

I have proven the converse, and I know that if left Kan extensions existed and were absolute, i.e. preserved by any functor $G: E \to E'$, then $Ran_{F}id_C$ would be the desired adjoint.

I initially looked for a counterexample, but this turned out to be unsuccessful. I really have no idea how to approach this unfortunately, so any help is appreciated!

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  • $\begingroup$ @Max maybe this is something you are interested in or can help with? $\endgroup$ – ThePuix Oct 6 '19 at 17:58
  • $\begingroup$ Maybe this can help : math.stackexchange.com/questions/2314042/… . I don't know the answer at the moment, although note : if $i$ is in fact a left adjoint (= has a right adjoint), then the Kan extensions $Lan_i F$ are all absolute. In other words, if $i$ has all Kan extensions along it, but some of them aren't absolute, then $i$ can't be a left adjoint. (1/2) $\endgroup$ – Maxime Ramzi Oct 6 '19 at 18:33
  • $\begingroup$ So two possible strategies are: find an example with non absolute extensions; or prove that with your hypotheses all extensions are absolute (specifically, you only need $Lan_i id_C$ to be absolute). [unrelated : I don't know if you @ was to me but I didn't get a notification, I just stumbled onto your question by chance ] (2/2) $\endgroup$ – Maxime Ramzi Oct 6 '19 at 18:35
  • $\begingroup$ @Max Was trying to get a hold of you, but I guess it doesn't work like that. And thanks for the possible strategies :). Do you have any intuition for what the answer might be? $\endgroup$ – ThePuix Oct 6 '19 at 19:14
  • $\begingroup$ My intuition about Kan extensions is not very good yet, but I would say that only imposing existence will not yield absoluteness, and so I would say there are counterexamples. But maybe some trick with some functor or comma category can yield the desired properties, so I'm not quite sure. $\endgroup$ – Maxime Ramzi Oct 6 '19 at 20:05

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