So considering a Monad to be a Triple (T:C -> C, η, µ) with eta and mu as the Natural transformations with appropriate signatures, isn't this in essence an F-Algebra?

My thinking is that being both (Monad and F-Alg) centered around some particular action(s) on|of a given Endofunctor, what stops Monads NOT just being special cases of F-Algebra's? (Also for some context I've been reading a bit of Homotopy theory & Bart Jacobs so I like could just be you know not seeing clearly and making connections where there are none...) I'm seeing a Monad as an F-Algebra of some Monoidal category or Quasicategory with the action of the Endofunctor or alternatively the operations definable on the Algebra of the Endofunctor as the particular semantics of that F-Algebra (e.g. what makes a Monad the State Monad or the Identity Monad or what have you instead...)

Examples, Proofs etc. in Scala or Haskell are good, but I might need clarification if it's in Haskell.


The paper with a rather suggestive title "Algebras, Coalgebras, Monads and Comonads" (2001) by Neil Ghani, Christoph Lüth, Federico De Marchi, John Power addresses this topic.

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  • $\begingroup$ I'm gonna leave this open for a while (a few days at least) but I appreciate the answer Dave, and I certainly accept it, but not yet... $\endgroup$ – Mzk Levi Jun 22 '13 at 22:23
  • $\begingroup$ That's understandable. Someone might be able to provide you with more details. $\endgroup$ – Dave Clarke Jun 23 '13 at 4:50

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