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This is a question about a naming convention. The Barr-Beck theorem (or simply Barr-Beck) is used a lot in descent theory over the past 30 years, almost invariably without a reference, like folklore.

To make precise which theorem I am talking about: According to one source the Barr-Beck monadicity theorem gives necessary and sufficient conditions for a category to be equivalent to a category of algebras over a monad.

There are occasionally references, e.g., to the Wikipedia page on "Beck's monadicity theorem" which attributes the theorem to Beck's thesis in 1967 under the direction of Eilenberg.

This makes me wonder how Barr is related to Barr-Beck. So I did some further research and found out that there is indeed a Barr-Beck paper with a Barr-Beck theorem. But it is about triple cohomology of algebras. This seems to be different. So here are my questions:

Is the Barr-Beck theorem different from Beck's theorem? If so, how? If not, how did Barr's name get attached to it?

By the way, the second Beck is Jon Beck while the first is Jonathan Mock Beck with a different author number in MathSciNet. So out of curiosity: Are Jon and Jonathan Beck one and the same person?

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    $\begingroup$ If I remember correctly you can find the theorem in M. Barr, Ch. Wells, “Toposes, triples and theories”, Repr. Theory Appl. Categ., 12 (2005), 1–288. $\endgroup$ – Sasha Feb 17 at 16:25
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    $\begingroup$ They must be the same mathematician: the author "Beck, Jon" (with Barr) of "Acyclic models and triples", 1966 must be the same as the author "Beck, Jonathan Mock" of TRIPLES, ALGEBRAS AND COHOMOLOGY, Thesis 1967. Wikipedia agrees, see article on "Jonathan Mock Beck (aka Jon Beck)". $\endgroup$ – Balazs Feb 17 at 16:42
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    $\begingroup$ Recall that "triple" is just the old-timey way to say "monad". I don't know why Barr's name specifically is attached, but another great thing about Toposes, triples, and theories mentioned by Sasha above is that they prove maybe half a dozen different versions of the monadicity theorem. Beck's original version can be hard to use, sort of like the General Adjoint Functor Theorem -- the "special" versions have more restrictive, but easier-to-check, hypotheses. Maybe one particular version of the theorem is Barr's. $\endgroup$ – Tim Campion Feb 17 at 16:43
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    $\begingroup$ Just to confirm that they are indeed the same person, and the thesis has been re-typeset and disseminated as tac.mta.ca/tac/reprints/articles/2/tr2abs.html $\endgroup$ – Yemon Choi Feb 17 at 19:21
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    $\begingroup$ Thanks. I informed the editors of MathSciNet. $\endgroup$ – Friedrich Knop Feb 17 at 19:31
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It is well-attested in the category theory literature (e.g. in Mac Lane's Categories for the Working Mathematician, Chapter VI) that the well-known theorem giving necessary and sufficient conditions for monadicity of a functor is due to Jon Beck. Indeed, most category theorists I know call this "Beck's monadicity theorem".

So why do others so often call it the "Barr-Beck theorem" nowadays?

As Tim points out in the comments above, there are many variants of the monadicity theorem which give sufficient, but not generally necessary, conditions for a functor to be monadic. Several of these variants may be found in the Exercises to Section VI.7 of Mac Lane (which section is titled "Beck's Theorem"). Exercise 7 reads as follows:

  1. CTT (Crude Tripleability Theorem; Barr-Beck). If $G$ is CTT, prove that the comparison functor $K$ is an equivalence of categories.

(A functor $G \colon A \to B$ is said by Mac Lane to be CTT when it is has a left adjoint, it preserves and reflects all coequalizers which exist, and the category $A$ has coequalizers of all parallel pairs of arrows $f,g$ such that the pair $Gf,Gg$ has a coequalizer in $B$. Note that these conditions are much stronger than those appearing in Beck's monadicity theorem. Indeed, the composite of two CTT functors is CTT, whereas in general the composite of two monadic functors is need not be monadic.)

Note the attribution to Barr-Beck. Now, this particular theorem was cited by Deligne in Section 4.1 of his paper Catégories tannakiennes in the Grothendieck Festschrift as "Le théorème de Barr-Beck".

My theory is that the name "Barr-Beck theorem" was popularised in certain circles by Deligne's usage here, and that over time (how long?) its usage shifted in these circles to refer to Beck's precise monadicity theorem. I fear that this incorrect usage has now been set in stone by its appearance in the works of modern influential authors such as Lurie.

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    $\begingroup$ Incidentally, the precise form of Beck's monadicity theorem does not appear in Beck's thesis, and I don't think he published it anywhere else. Only a weaker form (again, giving sufficient but not necessary conditions) appears in the thesis. According to Mac Lane (in the Notes at the end of Chapter VI), Beck presented the precise form at a conference in 1966. $\endgroup$ – Alexander Campbell Feb 17 at 18:28
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    $\begingroup$ I wonder if Mac Lane's attribution of CTT to Barr-Beck has a similar backstory living outside of published references? Somebody should probably ask Michael Barr. He used to be on MO occasionally, but apparently he hasn't been on in some years. $\endgroup$ – Tim Campion Feb 17 at 18:33
  • $\begingroup$ I wonder that too. I asked Ross Street about this once but he said it was before his time! $\endgroup$ – Alexander Campbell Feb 17 at 18:39
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    $\begingroup$ Thanks! So Barr contributed to Becks's theorem after all by making it more manageable. $\endgroup$ – Friedrich Knop Feb 17 at 19:18
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It looks like Alexander's answer hits the nail on the head, but I thought I might add what Barr himself says. Barr and Wells write in Toposes, Triples, and Theories (on page 121 in the "Historical Notes on Triples" section):

The next important step was the tripleableness theorem of Beck’s which in part was a generalization of Linton’s results. Variations on that theorem followed (Duskin [1969], Paré [1971]) and acquired arcane names, but they all go back to Beck and Linton. They mostly arose either because of the failure of tripleableness to be transitive or because of certain special conditions.

The "arcane names" appears to be a reference to the "vulgar tripleability theorem" and the "crude tripleability theorem" which appear on page 108.

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M. Barr and J. Beck, Acyclic models and triples, Proc. Conference Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 336-343.

The "triple" is an older name for a monad.

Also, Jonathan Mock Beck is the same as Jon Beck, as you can see from these lecture notes and this Wikipedia page.


Follow-up: In Beck's thesis the monadicity theorem is discussed on page 8 with reference to an unpublished note: J. Beck, The tripleableness theorems. To appear. The editor's note adds that "To our knowledge, this has not appeared. Beck’s tripleableness theorems have been exposed in M. Barr and C. Wells, Toposes, Triples and Theories."
Beck refers in the intro of his thesis to the 1966 Barr & Beck paper cited above, saying "That paper contains a summary of the present work." This might explain the later attribution to Barr & Beck jointly.

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    $\begingroup$ No monadicity theorem appears in this paper. $\endgroup$ – Alexander Campbell Feb 17 at 17:45
  • $\begingroup$ This is the same paper that was mentioned in the question, right? $\endgroup$ – R. van Dobben de Bruyn Feb 17 at 18:00

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