A list is a monad

So I come at this from a math background but I’ve always found these explanations to be overly complex. In the parlance of C++, I think of a monad as a template class T with the following properties:

1. For any class X, there is a canonical method

  F: X -> T<X>

2. For any class X, there is a canonical method

  G: T<T<X>> -> T<X>.

3. For classes X and Y, and any method

  f: X -> Y, 

there is a corresponding method

  “T<f>”: T<X> -> T<Y>.

—————-

Here “any type” means any type that is compatible with the template.

And then there’s some additional rules which make all these methods compatible, based on the different ways of stacking nested T’s and the “canonical” maps you get. Admittedly there is some confusing accounting here, but I also think most natural ways of constructing the above three requirements are going to satisfy them anyway. For List and Maybe it’s fairly obvious what the above methods are.

I dunno, maybe I have it wrong and someone can correct my understanding.

So you think that a monad which is an object with a simple definition in category theory is better explained in terms of C++?

I would agree that most of these articles about monads are bad. Just study the definition, then study what you can do with monads, it's not that hard.

Yes.

If you don't already know category theory, learning it is hard. The terms on wikipedia seem to form a dense graph of links. It's hard to get a foothold of comprehension. For people that already know C++, or are at least familiar with this syntax, this is more useful than describing it in haskell syntax or category theory. There seems to be a chicken and egg problem regarding haskell and monads. Learning c++ may be harder or easier than category theory. I'm no sure, as I don't understand either one of them. But this syntax makes more sense to me than something expressed in terms of category theory vocabulary.

Learning category theory to the level of understanding monads should take half an hour at most, and would constitute real understanding of what a monad is, versus this C++ explanation which is handwavy even in terms of C++. C++ can't even encode monads accurately!

But one doesn't even need to learn category theory. I assume that everybody has learned abstract algebra in high school, monoid, rings, groups, vector spaces and all that. A monad is just another kind of a structure like that. If you have studied abstract algebra in school then it should take 5 seconds to read the definition of a monad, a minute to understand it, and perhaps 10 minutes to see how various things such as errors or lists form monads.

Learning category theory, or indeed any sort of math from Wikipedia is an absolute futile endeavour.