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Commutative algebraic theories were introduced by Linton in the 1966 paper Autonomous Equational Categories. Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal closed categories.

The correspondence between (finitary) algebraic theories and (finitary) monad is known to specialise to a correspondence between (finitary) commutative algebraic theories and (finitary) commutative monads. An explicit reference is Corollary 10.6 of Lucyshyn-Wright's Commutants for enriched algebraic theories and monads. However, this result must have been known much earlier. Though Kock cites Linton's paper, I could find no reference to commutative algebraic theories in any of his papers on commutative monads. An exercise in Wraith's 1970 lecture notes on algebraic theories suggests the connection was known, but it is not explicitly stated. A comment at the end of Lindner's paper Commutative monads suggests that, at least in 1975, it was conjectured the two were equivalent, but potentially had been not proven.

Where does a proof of the correspondence between commutative algebraic theories and commutative monads (finitary or infinitary) first appear? (If it is asserted somewhere, but not proven, I would also be interested to know.)

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    $\begingroup$ You're now probably as expert on the 1970s categorical literature as anyone else apart from its actual authors. Anders Kock proved the equivalence between monoidal and commutative monads. He's the person I would ask about the rest of your question, given that Fred Linton is no longer with us. $\endgroup$
    – Paul Taylor
    Commented Nov 28, 2021 at 20:28
  • $\begingroup$ Of course, particularly since you're in Cambridge, you could start by asking Peter Johnstone, who has (had) an extensive card index of categorical facts and literature. $\endgroup$
    – Paul Taylor
    Commented Nov 28, 2021 at 21:00
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    $\begingroup$ I emailed Anders Kock and he was unaware of an explicit reference, though he said it was known even as early as the late 1960s. $\endgroup$
    – varkor
    Commented Dec 1, 2021 at 15:45
  • $\begingroup$ A friend of mine told me it was even the 40s. ;) $\endgroup$
    – Ivan Di Liberti
    Commented Dec 8, 2021 at 17:56

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