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Many papers refer to an untitled manuscript of Jon Beck (Cornell, 1966) for the origin of the monadicity theorem (originally called a "tripleability theorem"). An early proof is in Manes's 1967 thesis A Triple Miscellany: Some Aspects of the Theory of Algebras over a Triple (Theorem 1.2.9). Manes cites Beck's 1967 thesis Triples, Algebras, and Cohomology as a reference, but the monadicity theorem does not actually appear there.

Where can one find a copy (preferably digitised) of the untitled manuscript of Beck containing the monadicity theorem? (Considering that the manuscript is cited, presumably a copy exists and was circulated, rather than passed on by word of mouth.)

Evidence for the existence of the manuscript is given by an email of Marta Bunge on the categories mailing list (dated 4th November 2007):

There is an unpublished (untitled and undated) four-pages manuscript which John Beck gave to me (and I supposed also to many ohers) when he was at McGill. In it, he states and proves two theorems, the CTT (crude tripleableness theorem), and the PTT (precise tripleableness theorem). There is a connection between triples and descent implicit in the PTT. But this is not the same connection with descent as the Benabou-Roubaud theorem.

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    $\begingroup$ I am a big fan of this MO original series "Varkor à la recherche de la référence perdue". $\endgroup$
    – Ivan Di Liberti
    Aug 14, 2021 at 22:12
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    $\begingroup$ You might try writing to Michael Barr. I don't know if he has a copy, but he seems a likely candidate. $\endgroup$
    – Alexander Campbell
    Aug 15, 2021 at 0:47
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    $\begingroup$ Thanks, I shall do so next. I asked Robert Paré and he said that he no longer has a copy; I agree Barr seems a likely candidate. $\endgroup$
    – varkor
    Aug 15, 2021 at 20:29
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    $\begingroup$ Michael Barr said he was unaware of the manuscript's existence. I shall continue asking academics who seem like they may know. $\endgroup$
    – varkor
    Aug 15, 2021 at 21:19

2 Answers 2

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After reaching out to every researcher who cited the manuscript, John Kennison was kind enough to find and scan his copy of the untitled manuscript containing the crude and precise monadicity theorems. I have uploaded it to the nLab for posterity: Jon Beck's untitled manuscript. This copy was distributed at the Conference Held at the Seattle Research Center of the Battelle Memorial Institute in June – July 1968, though evidence from citations suggests it was first distributed as early as 1966.

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    $\begingroup$ That's amazing, Nathanael! Your annoying questions have really paid off there! Also, Jon Beck had very clear handwriting! $\endgroup$
    – Paul Taylor
    Sep 16, 2021 at 18:32
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I checked the TAC reprint of Beck's Triples, Algebras, and Cohomology from 1967. It is evident from the discussion at pag. 8, before Thm. 1 that tripleability was not presented in writing by Beck before his thesis (1967). There, Beck promises a paper "to appear", whose intended title was The tripleableness theorems. The TAC reprint has an editor's note in the bibliography commenting on this paper.

Editors’ note: To our knowledge, this has not appeared. Beck’s tripleableness theorems have been exposed in M. Barr and C. Wells, Toposes, Triples and Theories. Springer-Verlag, Berlin, Heidelberg, New York, 1984 as well as other places.

My impression is thus that Beck was writing the paper while working on his thesis in 1966. When people refer to this forgotten manuscript they are actually referring to the discussion before Thm 1, a draft that maybe has circulated in a very sketchy form among very selected people and was probably never finished.

I also checked the papers that cite this The tripleableness theorems and judging from the 3 citations of Paré, he might have seen the manuscript. Barr cites it once too.

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    $\begingroup$ On the contrary, Mac Lane tells us (in the Notes to CWM, Chapter VI) that Beck presented the precise form of the monadicity theorem at a conference in 1966. Also, in the preface to the TAC reprint of Beck's thesis, we read: "Although the date on the thesis is 1967, there was a nearly complete draft circulated in 1964." $\endgroup$
    – Alexander Campbell
    Aug 15, 2021 at 7:41
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    $\begingroup$ @AlexanderCampbell I meant presented in a citable way. I can claim I proved anything at a talk. This is literally how our community became hyper-toxic and folklore-based in the 1970s. I am beyond that, papers of did not happen ;). $\endgroup$
    – Ivan Di Liberti
    Aug 15, 2021 at 7:48
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    $\begingroup$ @MartinBrandenburg: I wouldn’t quite say “hyper-toxic” myself, but there was a period — my impression is it was highest in the late 90’s and early 2000’s — where category theory was notoriously unwelcoming to young researchers, with their contributions dismissed by the old guard as “yes, we knew that already in the 70’s”, since so much more had been known and discussed in that period than was ever written up. I entered the field at the tail end of that period; and in the time I’ve known the field it’s become much more welcoming. $\endgroup$
    – Peter LeFanu Lumsdaine
    Aug 15, 2021 at 10:23
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    $\begingroup$ @Martin there were people publicly debating attribution rights for ideas/theorems/definitions in the mid to late 00s, based on discussions at a conference in the 60s. Because CT had no journal, pure CT was mostly published in Lecture Notes in Mathematics, when it got formally published at all. A lot was just presented at seminars, and a few people were notorious for only presenting work this way. And there are papers in bibliographies cited as to appear in a specific journal, as if they had been accepted for publication, which never turned up. Then no one else could publish that result. $\endgroup$
    – David Roberts ♦
    Aug 15, 2021 at 21:32
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    $\begingroup$ The series Reprints in Theory & Applications of Categories exists to remedy the problem that Peter and David describe. It (re)publishes important works that were never published or are hard to get hold of. If you know of something that deserves to be published in Reprints and meets the criteria (see tac.mta.ca/tac/reprints/geninfo.html), drop me or any other TAC editor an email. I don't know whether Beck's note would be a good fit - I've never seen it. $\endgroup$
    – Tom Leinster
    Aug 21, 2021 at 12:23

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