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?