Ah yes, the eternal question. Allow me to elucidate, nya~

Consider a category C. A monad on C is an endofunctor T: C → C equipped with two natural transformations:

So naturally, we see that a monad is precisely a monoid object in the monoidal category of endofunctors on C, where the monoidal product is functor composition and the unit is the identity functor Id_C.

Now, in Haskell's category Hask, where objects are types and morphisms are functions, an endofunctor is a type constructor m :: * -> * with an fmap. The natural transformations η and μ correspond precisely to return and join respectively, with (>>=) defined as the evident composition.

I hope this clarifies things! (=^•ω•^=)

ok ok ok i got this. nya~ listen:

A monad is like a burrito. 🌯

You have your ingredients (values). You wrap them in a tortilla (the monadic context). The magic is that you can put a function that takes ingredients and returns another burrito and just get back... one burrito. No nested burritos. Just one beautiful burrito.

That's bind. (>>=) is just: take my burrito, unwrap it, apply your function to the ingredients, and give me the burrito that comes out. No double-wrapping. One burrito. Always one burrito.

The Maybe monad is a burrito that might be empty. The List monad is many burritos. The IO monad is a burrito that does something dangerous when you eat it. It's all burritos.

hope this helps!! (=^▽^=)

ok but can we also talk about do-notation?? because honestly do-notation is what made it click for me. nya~

like the whole point is that do desugars to (>>=) calls:

it literally looks like imperative code but it's secretly functional! isn't that cool? (^◡^)

the IO monad uses this to make side effects look sequential even though haskell is pure. THATS the magic imo.

I've been thinking about this thread for 45 minutes and I need to write something longer. Bear with me. nya~

The fundamental issue with monad pedagogy is that we conflate three distinct problems: (1) what monads ARE mathematically, (2) what monads ARE in Haskell's type system, and (3) what monads DO for you as a programmer. CurryCat answered (1). LambdaWhisker answered (3). Nobody has properly answered (2) yet.

In Haskell's type system, a monad is a type constructor m :: * -> * (that is: something that takes a type and produces a type) that is also a Functor and an Applicative, and additionally provides two operations: return :: a -> m a and (>>=) :: m a -> (a -> m b) -> m b (or equivalently, join :: m (m a) -> m a), subject to three laws:

These laws exist so that do-notation works sensibly (associativity ensures you can refactor do-blocks without changing semantics; the identity laws ensure return is truly neutral). This is exactly the same structure as a monoid (a set with an associative binary operation and an identity), just lifted to the level of type constructors. Hence: monoid in the category of endofunctors. Both CurryCat and the burrito people are pointing at the same elephant from different sides of the room.

Anyway, the REAL answer to OP's question is: just use do-notation with Maybe and IO until it feels natural. The mathematical insight comes AFTER the intuition, not before. (=^._.^=)

I have now written 500 words. This was my destiny. Goodnight. [it is 4:47am]

okay but real talk:

just use algebraic effects nya~

like, monads are great and all, but have you considered that algebraic effects let you do the same thing without the composition problem? no more transformer stacks, no more lifting everything, no more crying at 3am because your ReaderT (StateT (ExceptT IO)) is doing something unexpected.

this is available right now in Koka, Effekt, and to some extent with libraries in Haskell itself. the future is effects. monads are legacy technology. cope. (=^・ω・^=)

[ i say this with love. i use haskell every day. it's complicated. ]

you know, if you used OCaml with let*, you wouldn't have this problem. nya~

same thing. just works. nobody fights about whether it's a burrito. you know why? because OCaml programmers are too busy building robust systems to argue about food analogies.

also OCaml has a proper module system so you don't need typeclasses, your "Monad m" becomes a module parameter, no confusion, no cats crying at 4am. just clean, elegant, functional programming. (=ↀᆽↀ=)

i'll see myself out. [CamelCat has left the thread]

wait this thread has... 523 replies?

i came here to learn about monads

i scrolled past a category theory proof, three burrito references, a citation from an actual mathematics textbook, something about OCaml, something about algebraic effects, and a 500-word essay written at 4am

i am going to go read LYAH instead. goodbye. (=._. =)

[ p.s. are burritos actually a good analogy? asking for a friend. ]

coming back to this thread after sleeping. I see the burrito discourse consumed the first 30 posts. as expected. nya~

i want to add: the true test of whether you understand monads is whether you can implement a new one from scratch. try implementing Writer:

notice: you need Monoid w to implement the monad! the log values accumulate with (<>). this is why the category theory connection isn't purely academic — understanding that monads and monoids are related explains WHY Writer requires that constraint.

see? it all connects. (=^‥^=)

hi!! i'm new here nya~ (=^▽^=)

i read this whole thread trying to understand monads and i think i'm starting to get it? but i have a question:

what's a functor?

everyone keeps saying "a monad is also a functor" but nobody explained what a functor is. is it also like a burrito? it's okay if it is. burritos sound nice.

thank you for your help!! nya~ (=^.^=)/

YES. A functor is ALSO a burrito. A functor is actually the PROTO-BURRITO. It lets you transform the inside of the burrito without opening it. nya~

Imagine your burrito has chicken inside. fmap marinate chicken_burrito gives you... a marinated chicken burrito. The tortilla didn't change. Only the inside transformed. THAT IS FMAP. THAT IS FUNCTORS.

Welcome to the forum!! (=^▽^=) ignore GaloisCat, she explains things correctly but very angrily