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This is my first post here, as someone from Mathsstack suggested this might me a more suitable forum for this specific question.

I have been reading some texts by Joaquim Lambek on formal languages, grammar and so on, and at some point he mentions that the kind of approach he advocates in the late 90s and 2000s has the advantage, as compared to some similar ones from the end 60s, that they allow for double adjoints, which, in turn, account for what other schools address with the use of traces.

Does that mean that double adjoints were ony developed in logic and mathematics in the 70s or later? Or how should I understand that claim? I am assuming that the authors in the 60s were not simply oblivious

I would be very grateful if you could me illustrate me on the history of double adjoints, with references, milestones, etc...

Thanks in advance.

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    $\begingroup$ Crossposted from math.stackexchange.com/questions/2544546/…. $\endgroup$
    – Noah Schweber
    Commented Nov 30, 2017 at 15:42
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    $\begingroup$ crossposted after one hour $\endgroup$
    – YCor
    Commented Dec 1, 2017 at 6:21
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    $\begingroup$ If you want people to answer your question, it's kind to them to include definitions of your terminology in the question, especially if it's nonstandard. $\endgroup$
    – Mike Shulman
    Commented Dec 1, 2017 at 21:16
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    $\begingroup$ The link is 15 pages long, and searching for the word "double" turns up only one occurrence on page 7, which is not a definition. $\endgroup$
    – Mike Shulman
    Commented Dec 3, 2017 at 3:41
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    $\begingroup$ Personal communication from Michael Barr: there is really no theory or history of double adjoints. Just that in some cases, a right (say) adjoint to a functor has a further right adjoint. The most obvious example of this is that a function $f\colon X \to Y$ induces the function direct image on the power sets (partially ordered sets considered as categories) $f_*\colon P(X) \to P(Y)$. (cont) $\endgroup$
    – peter a g
    Commented Dec 4, 2017 at 12:49

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