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There is an unpublished (untitled and undated) four-pages manuscript which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.
Best Answer
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.