I owe so much to this blog! It was such an inspiring read when I started out programming in Haskell back in 2007.
Today I am a professor in computer science and still draw on it for examples in my advanced functional programming course. Just last week we did the loeb function, as an example of interesting use of Functor.
I attended a talk of his at Papers We Love at Strange Loop in 2018, I didn't really read the description and I was vaguely expecting something Haskell related, and instead got this: https://www.youtube.com/watch?v=766obijdpuU
I could barely understand it, but was impressed by what I could grasp. Dan Piponi's range is amazing, dude is brilliant
Free Monads are everywhere. Learning Haskell at this time was such an amazing experience. Haskell has incredible library stability, the kmettoverse feels the same, my is still good enough for most situations, there are new streaming libraries but accomplish the same things as conduit and pipes. LLMs are as decent as you would expect on Haskell, and have helped me debug some situations where I would be fighting GHC usually with some flags turned out. AI has actually has been helpful in learning since in Haskell once you figure something out you solve it for a whole class of problems, the issues is sometimes figuring that one thing out it's so abstract you feel like you are hitting a cliff. Excited to be writing Haskell still in 2026, I hope it continues to avoid success at all cost.
People think category theory is weird and confusing, but really it just managed to name things (classes) that before were just "things". One might not know what monad or functor is, but they surely used it and have intuition on how it works.
Right. I don't know how many times I've been exasperated by how monads are perceived as difficult.
Do you understand "flatmap"? Good, that's literally all a monad is: a flatmappable.
Technically it's also an applicative functor, but at the end of the day, that gives us a few trivial things:
- a constructor (i.e., a way to put something inside your monad, exactly how `[1]` constructs a list out of a natural number)
- map (everyone understands this bc we use them with lists constantly)
- ap, which is basically just "map for things with more than one parameter"
Monads are easy. But when you tell someone "well it's a box and you can unwrap it and modify things with a function that also returns a box, and you unwrap that box take the thing out and put it inside the original box—
No. It is a flatmappable. That's it. Can you flatmap a list? Good. Then you already can use the entirety of monad-specific properties.
When you start talking about Maybe, Either, etc. then you've moved from explaining monads to explaining something else.
It's like saying "classes are easy" and then someone says "yeah well what about InterfaceOrienterMethodContainerArrangeableFilterableClass::filter" that's not a class! That's one method in a specific class. Not knowing it doesn't mean you don't understand classes. It just means you don't have the standard library memorized!
> everyday business and physics is monadic in function.
So?
> And if-then statements are functorial.
So?
All the "this is hard" stuff around these ideas seems to focus on managing to explain what these things are but I found that to progress at the speed of reading (so, about as easy as anything can be) once it occurred to me to find explanations that used examples in languages I was familiar with, instead of Haskell or Haskell-inspired pseudocode.
What I came out the other side of this with was: OK, I see what these are (that's incredibly simple, it turns out) and I even see how these ideas would be useful in Haskell and some similar languages, because they solve problems with and help one communicate about problems particular to those languages. I do not see why it matters for... anything else, unless I were to go out of my way to find reasons to apply these ideas (and why would I do that? And no, I don't find "to make your code more purely-functional" a compelling reason, I'm entirely fine with code I touch only selectively, sometimes engaging with or in any of that sort of thing).
It's also important to note that in Haskell and other functional programming languages, there is no implied order of operations. You need a Monad type in order to express that certain things are supposed to happen after other things. Monads can also express that certain things happen "in between" two operations, which is why we have different kinds of Monads and mathematical axioms of what they're all supposed to do.
Outside of FP however, this seems really stupid. We're used to operations that happen in the order you wrote them in and function applications that just so happen to also print things to the screen or send bits across the network. If you live in this world, like most people do, then "flatmap" is a good metaphor for Monads because that's basically all they do in an imperative language[1].
Well, that, and async code. JavaScript decided to standardize on a Monad-shaped "thenable" specification for representing asynchronous processes, where most other programming languages would have gone with green threads or some other software-transparent async mechanism. To be clear, it's better than the callback soup you'd normally have[0], but working with bare Thenables is still painful. Just like working with bare Monads - which is why Haskell and JavaScript both have syntax to work around them (await/async, do, etc).
Maybe/Either get talked about because they're the simplest Monads you can make, but it makes Monads sound like a spicy container type.
[0] The FP people call this "continuation-passing style"
[1] To be clear, Monads don't have to be list-shaped and most Monads aren't.
There is an implied order of operations in Haskell. Haskell always reduces to weak head normal form. This implies an ordering.
Monads have nothing to do with order (they follow the same ordering as Haskell's normalization guarantees).
> JavaScript decided to standardize on a Monad-shaped "thenable" specification for representing asynchronous processes,
Its impossible for something to be monad shaped. All asynchronous interfaces form a monad whether you decide to follow the Haskell monad type class or decide to do something else. They're all isomorphic and form a monad. Any model of computation forms a monad.
Assembly language quite literally forms a category over the monoid of endo functors.
Jacquard loom programming also forms a category over the monoid of endo functors because all processes that sequence things with state form such a thing, whether you know that or not.
It's like claiming the Indians invented numbers to fit the addition algorithm. Putting the cart before the horse, because all formations of the natural numbers form a natural group/ring with addition and multiplication formed the standard way (they also all form separate groups and rings, that we barely ever use).
But this has absolutely nothing to do with "certain things are supposed to happen after other things", and CANNOT POSSIBLY have anything to do with that. Flatmap is a purely functional concept, and in the context of things that are purely functional, nothing ever actually happens. That's the whole point of "functional" as a concept. It cleanly separates the result of a computation from the process used to produce that result.
So one of your "simple" explanations must be wrong.
Because you're not used to abstract algebra. JavaScript arrays form a monad with flatmap as the join operator. There are multiple ways to make a monad with list like structures.
And you are correct. Monads have nothing to do with sequencing (I mean any more than any other non commutative operator -- remember x^2 is not the same as 2^x)
Haskell handles sequencing by reducing to weak head normal form which is controlled by case matching. There is no connection to monads in general. The IO monad uses case matching in its implementation of flatmap to achieve a sensible ordering.
As for JavaScript flat map, a.flatMap(b).flatMap(c) is the same as a.flatMap(function (x) { return b(x).flatMap(c);}).
This is the same as promises:
a.then(b).then(c) is the same as a.then(function (x) { return b(x).then(c)}).
Literally everything for which this is true forms a monad and the monad laws apply.
People have different "aha" moments with monads. For me, it was realizing that something being a monad has to do with the type/class fitting the monad laws. If the monad laws hold for the type/class then you've got a monad, otherwise not.
So then when you look at List, Maybe, Either, et al. it's interesting to see how their conforming to the laws "unpacks" differently with respect to what they each do differently (what's happening to the data in your program), but the laws are just the same.
The reason this was an aha moment for me is that I struggled with wanting to understand a monad as another kind of thing — "I understand what a function is, I understand what objects and primitive values are, but I don't get that List and Maybe and Either are the same kind of thing, they seem like totally different things!"
Yes, I 100% agree. But I want to mention something that isn't a disagreement, just a further nuance:
1. my explanation of monad is sufficient for people who need to use them
2. your explanation of monad is necessary for people who might want to invent new ones
What I mean by this is that if you want to invent a new monad, you need to make sure your idea conforms to the monad laws. But if you're just going to consume existing monads, you don't need to know this. You only need to know the functions to work with a monad: flatmap (or map + flatten), ap(ply), bind/of/just. Everything else is specific to a given monad. Like an either's toOptional is not monadic. It's just turning Left _ into None and Right an into Some a.
And needing to know these properties "work" is unnecessary, as their very existence in the library is pretty solid evidence that you can use them, haha.
> Do you understand "flatmap"? Good, that's literally all a monad is: a flatmappable.
Awesome! Now I understand.
> Technically it's also an applicative functor
Aaaand you've lost me. This is probably why people think monads are difficult. The explanations keep involving these unfamiliar terms and act like we need to already know them to understand monads. You say it's just a flatmappable, but then it's also this other thing that gives you more?
But words like "incapsulation" or "polymorphism" or even "autoincrement" also sound unfamiliar and scary to a young kid who encounters them the first time. But the kid learns their meaning along the way, in a desire to build their own a game, or something. The feeling that one already knows a lot, sort of enough, and it'd be painful and boring to learn another abstract thing is a grown-up problem :-\
I mean people need to be familiar with mathematics. In mathematics things form things without having to understand them.
For example, The natural numbers form a ring and field over normal addition and multiplication , but you don't need to know ring theory to add numbers..
People need to stop worrying about not understanding things. No one understands everything.
It's serendipitous that I'm seeing this blog post on the front page today, because I'm currently writing an article discussing the free monad.
In addition to the free monad presented in this post, there is a variant, called the "freer" monad, based on the "bind" operation instead of the "join" operation:
data Freer f a where
Pure :: a -> Freer f a
Bind :: f a -> (a -> Freer f b) -> Freer f b
When thinking of monads as giving the semantics of some computational strategy, it's easier to define them in terms of "bind" instead of "join." This way of defining monads is sometimes called a "Kleisli triple" because it is better suggestive of "Kleisli arrows," or functions of the signature `a -> m b`. The "bind" operation defines how to compose a monadic computation with its continuation, and from this perspective, the "freer" monad resembles an abstract syntax tree.
Originally, Eugenio Moggi proposed monads as a technique for specifying the denotational semantics of programming languages. All Java programs "really" happen in the IO + Either monads, because all Java programs may perform IO and throw exceptions. To my understanding, free monads are the monad that OCaml 5 runs in, because they give the semantics for effect handlers (or resumable exceptions).
> All Java programs "really" happen in the IO + Either monads
People say things like this all the time, and I think it's a vacuous assertion. While there is probably some very narrow view where returning early with an error, throwing an exception, and binding in Either are the same thing, such a view ignores a lot of important context (e.g. all of imperative programming). This is why you have to qualify it as "IO + Either", but that doesn't say anything at all because everything is possible in IO.
I keep bouncing off this stuff due to the lack of concrete examples where it would be useful. Maybe some documentation organized like the Design Patterns book would be helpful?
A great way to understand monads is as a "design pattern," because they pop up extremely often in practical programming. Consider functions with a type signature that looks like `A -> T<B>`, such as `A -> Promise<B>` or `A -> Optional<B>`.
If you have some function fetchResponse that returns a Promise<Response>, and a function processJSON that takes a Response as an argument, you cannot compose them the usual way, as `processJSON(fetchResponse(x))`. Instead, you need to do `fetchReponse(x).then(processJSON)`, and the entire expression has to return another Promise. Ditto for functions that return an Optional.
All data types that implement this design pattern have the structure of a monad. A monad consists of a generic type that is "covariant" in its type parameter (such as Promise or Optional), a way to embed a singular value into the data type, and a "then" method to compose the data type with a callback. Lists also implement the monad design pattern, and the "then" method for lists is flatmap. A monad basically lets you compose functions with a type signature that looks like `A -> T<B>`.
Furthermore, each of these data types (Promises, Optionals, Lists) can be viewed as the output of some computation. Promises are produced whenever a function performs asynchronous IO, Optionals are produced when computations may return some "null" value, and Lists are returned if an algorithm may produce multiple solutions.
Just like Promises have async/await syntactic sugar, similar syntactic sugar can be devised for other "monadic" types. For Optionals, the equivalent of async/await is null propagation (some languages have a `?` operator for this). For Lists, the equivalent of async/await is list comprehension, which "picks" each element from the list to build up a new list. Async/await, null propagation, and list comprehensions all have the same underlying structure, called a "monad."
A "free monad" is a monad that does not implement any specific computation, but instead builds up an abstract syntax tree that needs to be interpreted. Free monads are useful because other monad instances can be implemented in terms of the free monad.
I owe so much to this blog! It was such an inspiring read when I started out programming in Haskell back in 2007.
Today I am a professor in computer science and still draw on it for examples in my advanced functional programming course. Just last week we did the loeb function, as an example of interesting use of Functor.
Loeb function: http://blog.sigfpe.com/2006/11/from-l-theorem-to-spreadsheet...
Yeah this is such an important blog for Haskell.
A classic that everyone should read: http://blog.sigfpe.com/2006/08/you-could-have-invented-monad...
I attended a talk of his at Papers We Love at Strange Loop in 2018, I didn't really read the description and I was vaguely expecting something Haskell related, and instead got this: https://www.youtube.com/watch?v=766obijdpuU
I could barely understand it, but was impressed by what I could grasp. Dan Piponi's range is amazing, dude is brilliant
Free Monads are everywhere. Learning Haskell at this time was such an amazing experience. Haskell has incredible library stability, the kmettoverse feels the same, my is still good enough for most situations, there are new streaming libraries but accomplish the same things as conduit and pipes. LLMs are as decent as you would expect on Haskell, and have helped me debug some situations where I would be fighting GHC usually with some flags turned out. AI has actually has been helpful in learning since in Haskell once you figure something out you solve it for a whole class of problems, the issues is sometimes figuring that one thing out it's so abstract you feel like you are hitting a cliff. Excited to be writing Haskell still in 2026, I hope it continues to avoid success at all cost.
Monads, arrows, applicatives, and functors make the world go round.
People think category theory is weird and confusing, but really it just managed to name things (classes) that before were just "things". One might not know what monad or functor is, but they surely used it and have intuition on how it works.
Right. I don't know how many times I've been exasperated by how monads are perceived as difficult.
Do you understand "flatmap"? Good, that's literally all a monad is: a flatmappable.
Technically it's also an applicative functor, but at the end of the day, that gives us a few trivial things:
- a constructor (i.e., a way to put something inside your monad, exactly how `[1]` constructs a list out of a natural number)
- map (everyone understands this bc we use them with lists constantly)
- ap, which is basically just "map for things with more than one parameter"
Monads are easy. But when you tell someone "well it's a box and you can unwrap it and modify things with a function that also returns a box, and you unwrap that box take the thing out and put it inside the original box—
No. It is a flatmappable. That's it. Can you flatmap a list? Good. Then you already can use the entirety of monad-specific properties.
When you start talking about Maybe, Either, etc. then you've moved from explaining monads to explaining something else.
It's like saying "classes are easy" and then someone says "yeah well what about InterfaceOrienterMethodContainerArrangeableFilterableClass::filter" that's not a class! That's one method in a specific class. Not knowing it doesn't mean you don't understand classes. It just means you don't have the standard library memorized!
Forget programming, everyday business and physics is monadic in function.
And if-then statements are functorial.
These are very general thought patterns.
> everyday business and physics is monadic in function.
So?
> And if-then statements are functorial.
So?
All the "this is hard" stuff around these ideas seems to focus on managing to explain what these things are but I found that to progress at the speed of reading (so, about as easy as anything can be) once it occurred to me to find explanations that used examples in languages I was familiar with, instead of Haskell or Haskell-inspired pseudocode.
What I came out the other side of this with was: OK, I see what these are (that's incredibly simple, it turns out) and I even see how these ideas would be useful in Haskell and some similar languages, because they solve problems with and help one communicate about problems particular to those languages. I do not see why it matters for... anything else, unless I were to go out of my way to find reasons to apply these ideas (and why would I do that? And no, I don't find "to make your code more purely-functional" a compelling reason, I'm entirely fine with code I touch only selectively, sometimes engaging with or in any of that sort of thing).
The "so?" is the part I found (and find) hard.
These are vacuous statements.
It's also important to note that in Haskell and other functional programming languages, there is no implied order of operations. You need a Monad type in order to express that certain things are supposed to happen after other things. Monads can also express that certain things happen "in between" two operations, which is why we have different kinds of Monads and mathematical axioms of what they're all supposed to do.
Outside of FP however, this seems really stupid. We're used to operations that happen in the order you wrote them in and function applications that just so happen to also print things to the screen or send bits across the network. If you live in this world, like most people do, then "flatmap" is a good metaphor for Monads because that's basically all they do in an imperative language[1].
Well, that, and async code. JavaScript decided to standardize on a Monad-shaped "thenable" specification for representing asynchronous processes, where most other programming languages would have gone with green threads or some other software-transparent async mechanism. To be clear, it's better than the callback soup you'd normally have[0], but working with bare Thenables is still painful. Just like working with bare Monads - which is why Haskell and JavaScript both have syntax to work around them (await/async, do, etc).
Maybe/Either get talked about because they're the simplest Monads you can make, but it makes Monads sound like a spicy container type.
[0] The FP people call this "continuation-passing style"
[1] To be clear, Monads don't have to be list-shaped and most Monads aren't.
There is an implied order of operations in Haskell. Haskell always reduces to weak head normal form. This implies an ordering.
Monads have nothing to do with order (they follow the same ordering as Haskell's normalization guarantees).
> JavaScript decided to standardize on a Monad-shaped "thenable" specification for representing asynchronous processes,
Its impossible for something to be monad shaped. All asynchronous interfaces form a monad whether you decide to follow the Haskell monad type class or decide to do something else. They're all isomorphic and form a monad. Any model of computation forms a monad.
Assembly language quite literally forms a category over the monoid of endo functors.
Jacquard loom programming also forms a category over the monoid of endo functors because all processes that sequence things with state form such a thing, whether you know that or not.
It's like claiming the Indians invented numbers to fit the addition algorithm. Putting the cart before the horse, because all formations of the natural numbers form a natural group/ring with addition and multiplication formed the standard way (they also all form separate groups and rings, that we barely ever use).
> You need a Monad type in order to express that certain things are supposed to happen after other things
This is the kind of explanation that drives me absolutely batshit crazy because it is fundamentally at odds with:
> Do you understand "flatmap"? Good, that's literally all a monad is: a flatmappable.
So, I think I understand flatmap, assuming that this is what you mean:
https://www.w3schools.com/Jsref/jsref_array_flatmap.asp
But this has absolutely nothing to do with "certain things are supposed to happen after other things", and CANNOT POSSIBLY have anything to do with that. Flatmap is a purely functional concept, and in the context of things that are purely functional, nothing ever actually happens. That's the whole point of "functional" as a concept. It cleanly separates the result of a computation from the process used to produce that result.
So one of your "simple" explanations must be wrong.
Because you're not used to abstract algebra. JavaScript arrays form a monad with flatmap as the join operator. There are multiple ways to make a monad with list like structures.
And you are correct. Monads have nothing to do with sequencing (I mean any more than any other non commutative operator -- remember x^2 is not the same as 2^x)
Haskell handles sequencing by reducing to weak head normal form which is controlled by case matching. There is no connection to monads in general. The IO monad uses case matching in its implementation of flatmap to achieve a sensible ordering.
As for JavaScript flat map, a.flatMap(b).flatMap(c) is the same as a.flatMap(function (x) { return b(x).flatMap(c);}).
This is the same as promises: a.then(b).then(c) is the same as a.then(function (x) { return b(x).then(c)}).
Literally everything for which this is true forms a monad and the monad laws apply.
People have different "aha" moments with monads. For me, it was realizing that something being a monad has to do with the type/class fitting the monad laws. If the monad laws hold for the type/class then you've got a monad, otherwise not.
So then when you look at List, Maybe, Either, et al. it's interesting to see how their conforming to the laws "unpacks" differently with respect to what they each do differently (what's happening to the data in your program), but the laws are just the same.
The reason this was an aha moment for me is that I struggled with wanting to understand a monad as another kind of thing — "I understand what a function is, I understand what objects and primitive values are, but I don't get that List and Maybe and Either are the same kind of thing, they seem like totally different things!"
Yes, I 100% agree. But I want to mention something that isn't a disagreement, just a further nuance:
1. my explanation of monad is sufficient for people who need to use them
2. your explanation of monad is necessary for people who might want to invent new ones
What I mean by this is that if you want to invent a new monad, you need to make sure your idea conforms to the monad laws. But if you're just going to consume existing monads, you don't need to know this. You only need to know the functions to work with a monad: flatmap (or map + flatten), ap(ply), bind/of/just. Everything else is specific to a given monad. Like an either's toOptional is not monadic. It's just turning Left _ into None and Right an into Some a.
And needing to know these properties "work" is unnecessary, as their very existence in the library is pretty solid evidence that you can use them, haha.
> Do you understand "flatmap"? Good, that's literally all a monad is: a flatmappable.
Awesome! Now I understand.
> Technically it's also an applicative functor
Aaaand you've lost me. This is probably why people think monads are difficult. The explanations keep involving these unfamiliar terms and act like we need to already know them to understand monads. You say it's just a flatmappable, but then it's also this other thing that gives you more?
But words like "incapsulation" or "polymorphism" or even "autoincrement" also sound unfamiliar and scary to a young kid who encounters them the first time. But the kid learns their meaning along the way, in a desire to build their own a game, or something. The feeling that one already knows a lot, sort of enough, and it'd be painful and boring to learn another abstract thing is a grown-up problem :-\
Those words need definitions, but they can both be defined using words most people know.
Monad definitions often just sweep a pile of confusion around the room a lot.
I mean people need to be familiar with mathematics. In mathematics things form things without having to understand them.
For example, The natural numbers form a ring and field over normal addition and multiplication , but you don't need to know ring theory to add numbers..
People need to stop worrying about not understanding things. No one understands everything.
It's serendipitous that I'm seeing this blog post on the front page today, because I'm currently writing an article discussing the free monad.
In addition to the free monad presented in this post, there is a variant, called the "freer" monad, based on the "bind" operation instead of the "join" operation:
I believe this definition originates from the following paper by Oleg Kiselyov and Hiromi Ishii: https://okmij.org/ftp/Haskell/extensible/more.pdfWhen thinking of monads as giving the semantics of some computational strategy, it's easier to define them in terms of "bind" instead of "join." This way of defining monads is sometimes called a "Kleisli triple" because it is better suggestive of "Kleisli arrows," or functions of the signature `a -> m b`. The "bind" operation defines how to compose a monadic computation with its continuation, and from this perspective, the "freer" monad resembles an abstract syntax tree.
Originally, Eugenio Moggi proposed monads as a technique for specifying the denotational semantics of programming languages. All Java programs "really" happen in the IO + Either monads, because all Java programs may perform IO and throw exceptions. To my understanding, free monads are the monad that OCaml 5 runs in, because they give the semantics for effect handlers (or resumable exceptions).
> All Java programs "really" happen in the IO + Either monads
People say things like this all the time, and I think it's a vacuous assertion. While there is probably some very narrow view where returning early with an error, throwing an exception, and binding in Either are the same thing, such a view ignores a lot of important context (e.g. all of imperative programming). This is why you have to qualify it as "IO + Either", but that doesn't say anything at all because everything is possible in IO.
I keep bouncing off this stuff due to the lack of concrete examples where it would be useful. Maybe some documentation organized like the Design Patterns book would be helpful?
A great way to understand monads is as a "design pattern," because they pop up extremely often in practical programming. Consider functions with a type signature that looks like `A -> T<B>`, such as `A -> Promise<B>` or `A -> Optional<B>`.
If you have some function fetchResponse that returns a Promise<Response>, and a function processJSON that takes a Response as an argument, you cannot compose them the usual way, as `processJSON(fetchResponse(x))`. Instead, you need to do `fetchReponse(x).then(processJSON)`, and the entire expression has to return another Promise. Ditto for functions that return an Optional.
All data types that implement this design pattern have the structure of a monad. A monad consists of a generic type that is "covariant" in its type parameter (such as Promise or Optional), a way to embed a singular value into the data type, and a "then" method to compose the data type with a callback. Lists also implement the monad design pattern, and the "then" method for lists is flatmap. A monad basically lets you compose functions with a type signature that looks like `A -> T<B>`.
Furthermore, each of these data types (Promises, Optionals, Lists) can be viewed as the output of some computation. Promises are produced whenever a function performs asynchronous IO, Optionals are produced when computations may return some "null" value, and Lists are returned if an algorithm may produce multiple solutions.
Just like Promises have async/await syntactic sugar, similar syntactic sugar can be devised for other "monadic" types. For Optionals, the equivalent of async/await is null propagation (some languages have a `?` operator for this). For Lists, the equivalent of async/await is list comprehension, which "picks" each element from the list to build up a new list. Async/await, null propagation, and list comprehensions all have the same underlying structure, called a "monad."
A "free monad" is a monad that does not implement any specific computation, but instead builds up an abstract syntax tree that needs to be interpreted. Free monads are useful because other monad instances can be implemented in terms of the free monad.
https://jerf.org/iri/post/2025/fp_lessons_purity/ may help, perhaps even especially if you are not a functional programmer.
See also https://jerf.org/iri/post/2958/ .