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I am fine-tuning a short note on basic category theory; any such course must introduce monads, and I want to give a bit of history of the subject.

I soon realized that I don't know the precise series of events that led Mac Lane to create the name "monad" instead of the less creative "triple" or "standard construction". Did he coin the term, or did he simply advertize it?

The only reference I can find is a link at the nLab page.

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  • $\begingroup$ Surely a kind of back-formation from monoid? Coupled perhaps with the love of purloining terms from philosophy. I don't know that it needed more than one event, namely the recognition that a monad is a monoid in a category of endofunctors, like his book says. $\endgroup$
    – Todd Trimble
    Commented Aug 10, 2019 at 20:31
  • $\begingroup$ see also stackoverflow.com/q/3870088 (with an answer that has 754 upvotes...) $\endgroup$
    – Carlo Beenakker
    Commented Aug 10, 2019 at 20:35
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    $\begingroup$ My guess about the broken link is that it is to the email of Ross Street to the categories mailing list (dated 03 Nov 2007) starting with the sentence "History can be harder than mathematics.", available from mta.ca/~cat-dist/archive/2007/07-11 There was a lot of discussion about history in that email thread, and not 100% agreement on events from the 1960s (CT had a poor publication record at the time, due to the lack of dedicated journals, so results percolated via conference proceedings/abstracts, discussions etc. I think few people saw it then as a field of research) $\endgroup$
    – David Roberts ♦
    Commented Aug 11, 2019 at 2:00
  • $\begingroup$ I think that some authors use the term triple for a different presentation of monad. For these authors a triple is not a functor but an object functions with some additional morphisms parameterized by objects. Between these authors there are Moggi and Linton if I am not mistaken. $\endgroup$
    – Giorgio Mossa
    Commented Aug 11, 2019 at 12:38
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    $\begingroup$ You might be interested by this English.SE question $\endgroup$
    – Arnaud D.
    Commented Aug 12, 2019 at 6:52

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This is covered in this English.SE question. In short, people were not all very happy about the term "triple", and tried to come up with something better. Jean Bénabou suggested "monad" during lunch at a meeting in 1966, and it was quickly adopted; for example, it appears in the titles of Anders Kock's and Eduardo Dubuc's theses from 1967 and 1969 respectively, and the use of the term is alluded to in the Introduction of the "Seminar on triples and categorical homology theory" from 1969 and in Lawvere's "Ordinal sums and equational doctrines" from that volume. Peter May also convinced Mac Lane to use "monad" in his book, which helped to popularize the term.

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P. T. Johnstone (who wrote several books on Topos Theory) gave a Category Theory lecture in which he said this was originally called 'the standard construction', then 'triples', and finally 'monads' -- but he provided the caveat that mathematicians in Montreal still adhere to the term 'triples', such that if you're in Montreal and want to discuss monads, you should probably bear this in mind.

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    $\begingroup$ I'll bear this in mind if I go to Montreal. $\endgroup$
    – fosco
    Commented Aug 10, 2019 at 20:48
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    $\begingroup$ Notably Barr. (I'm not aware that everyone else is or was adamant about it, and I lived there for six months.) Added later: Joyal says "monad"; I've never heard him say "triple". $\endgroup$
    – Todd Trimble
    Commented Aug 10, 2019 at 21:09

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