<p>The paper with a rather suggestive title "<a href="http://www.sciencedirect.com/science/article/pii/S1571066104809058">Algebras, Coalgebras, Monads and Comonads</a>" (2001)
by Neil Ghani, Christoph Lüth, Federico De Marchi, John Power addresses this topic. </p>

CC BY-SA 3.0
2013-06-22T22:23:30.377Z
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...
Mzk Levi
CC BY-SA 3.0
2013-06-23T04:50:17.177Z
That's understandable. Someone might be able to provide you with more details.
Dave Clarke
CC BY-SA 3.0
2013-06-22T19:41:38.500Z
2013-06-22T19:41:38.500Z
Dave Clarke
<p>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?</p>

<p>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 &amp; 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...)</p>

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

<pl.programming-languages><haskell><monad><comonad>
Aren't Monads F-Algebra's? And then if that could be said are Comonad's F-Coalgebra's?
Mzk Levi
# Aren't Monads F-Algebra's? And then if that could be said are Comonad's F-Coalgebra's?

(Asked by: username_0 on 2013-06-22)

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.

**Tags:** pl.programming-languages, haskell, monad, comonad

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## 1 answers

### Answer by username_1 on 2013-06-22. Score: 5 | Accepted answer

The paper with a rather suggestive title "[Algebras, Coalgebras, Monads and Comonads](http://www.sciencedirect.com/science/article/pii/S1571066104809058)" (2001)
by Neil Ghani, Christoph Lüth, Federico De Marchi, John Power addresses this topic.

> **Comments:**
> **username_0**: 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...  
> **username_1**: That's understandable. Someone might be able to provide you with more details.  

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