A list is a monad

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kevinventullo ◴[02 Jul 25 23:40 UTC] No.44450020[source]▶

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.

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1. 4ad ◴[03 Jul 25 00:05 UTC] No.44450173[source]▶

>>44450020 #

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.

replies(2): >>44450531 #>>44451357 #

2. kevinventullo ◴[03 Jul 25 00:59 UTC] No.44450531[source]▶

>>44450173 (TP) #

For people who are more familiar with C++ than category theory, yes.

3. recursive ◴[03 Jul 25 03:25 UTC] No.44451357[source]▶

>>44450173 (TP) #

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.

replies(1): >>44454125 #

4. 4ad ◴[03 Jul 25 12:02 UTC] No.44454125[source]▶

>>44451357 #

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.

replies(2): >>44455412 #>>44455515 #

5. housecarpenter ◴[03 Jul 25 14:27 UTC] No.44455412{3}[source]▶

>>44454125 #

At least in my country (the UK), people generally do not learn abstract algebra in high school. That's a university-level topic.

I think there is a definite "step up" in complexity between the structures of abstract algebra such as monoids, rings, groups and vector spaces, and monads. All of those algebraic structures are basically just sets equipped with operations satisfying some equations. Monads are endofunctors equipped with natural transformations satisfying some equations. "Endofunctor" and "natural transformation" are considerably more advanced and abstract concepts than "set" and "operation", and they are concepts that belong to category theory (so I don't see how you can read and understand the definition of a monad without that basic level of category theory).

Your time estimates also seem wildly optimistic. A common rule of thumb is that reading a maths textbook at your level takes around an hour per page. I think the definition of a monad can be compared to one page of a textbook. So I'd say it'll take on the order of hours to read and understand the definition of a monad, and that's assuming you're already entirely comfortable with the pre-requisite concepts (natural transformations, etc.)

6. recursive ◴[03 Jul 25 14:37 UTC] No.44455515{3}[source]▶

>>44454125 #

I learned none of those in high school or university. I did indeed fail for much more than 30 minutes to learn any of them from Wikipedia and other sources.

Over the years I've spent many hours trying to make any sense of monads with varying degrees of success. But mostly not.

Your assertions do not seem consistent with reality as I've observed it.

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