The category of algebras over a monad (also: “modules over a monad”) is traditionally called its Eilenberg–Moore category (EM)

In that context

  1. What exactly does the word "Algebra" mean?
  2. What exactly does the expression "Algebra over..." mean?
  • 2
    $\begingroup$ You can find the definition of algebra over a monad in most of the category theory books. You can also find it online, e.g. en.wikipedia.org/wiki/… $\endgroup$
    – Luca Carai
    Nov 28, 2018 at 7:04
  • $\begingroup$ @LucaCarai I was looking for something like en.wikipedia.org/wiki/… $\endgroup$
    – Roland
    Nov 28, 2018 at 7:16
  • 2
    $\begingroup$ @Roland The word "algebra" is dramatically overloaded in mathematics. While there is some connection between the various uses, there is definitely no formal concept that they are all instances of. Also, for what it's worth, I usually just see "monad algebra" rather than "algebra over a monad" (though even more commonly I see $T$-algebra [versus $F$-algebra] where you're just supposed to know that $T$ stands for a monad [aka a triple]). $\endgroup$ Nov 28, 2018 at 7:33
  • 1
    $\begingroup$ The definition of algebra is given on the linked page, and there's also a link to another nLab page... $\endgroup$
    – Arnaud D.
    Nov 28, 2018 at 8:27


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