# Relationship between monoids and monads explained

### Monads are just monoids in the category of endofunctors

Functors, Applicatives, and Monads in Plain English - This is a no-bullshit zone. A million words of category theory and Haskell examples to say Swift doesn't have a built-in definition for applicative because you can't. In category theory, a branch of mathematics, a monad is an endofunctor together with two . Those axioms are formally similar to the monoid axioms. They are taken as the definition of a general monad (not assumed a priori to be connected to an .. interior algebras, and their relation to models of S4 and intuitionistic logics. A well-said, perhaps the briefest ever answer is: A monad is just a monoid in the category of endofunctors. A monoid is defined as an algebraic structure.

## Functors, Applicatives, and Monads in Plain English

Then I pass it all wrapped values, and I get a wrapped value out! There's a function called liftA2 that does the same thing: Get a PhD in computer science. Throw it away because you don't need it for this section! Monads add a new twist. Functors apply a function to a wrapped value: Applicatives apply a wrapped function to a wrapped value: Monads apply a function that returns a wrapped value to a wrapped value.

What is a Monad? - Computerphile

Let's see an example. Good ol' Maybe is a monad: Suppose half is a function that only works on even numbers: Here's how it works: Monad is another typeclass. Here's a partial definition: So Maybe is a Monad: And if you pass in a Nothing it's even simpler: You can also chain these calls: So now we know that Maybe is a Functor, an Applicative, and a Monad.

Now let's mosey on over to another example: By reifying a specific kind of computation, a monad not only encapsulates the tedious details of that computational pattern, but it does so in a declarative way, improving the code's clarity. As monads are also mathematically rigorous, they provide referential transparencyallowing for many techniques and optimizations based on rewriting.

A monad's general utility rather lies in simplifying a program's structure and improving separation of concerns through abstraction. Some monads can pass along extra data that is inaccessible to functions, and some even exert finer control over execution, for example only calling a function under certain conditions.

### Monad (functional programming) - Wikipedia

Because they let application programmers implement domain logic while offloading boilerplate code onto pre-developed modules, monads can even be considered a tool for aspect-oriented programming. Even purely functional languages can still implement these "impure" computations without monads, via an intricate mix of function composition and continuation-passing style CPS in particular.

When translated from category-theory to programming terms, the monad structure is a generic concept and can be defined directly in any language that supports an equivalent feature for bounded polymorphism. A concept's ability to remain agnostic about operational details while working on underlying types is powerful, but the unique features and stringent behavior of monads set them apart from other concepts.

In some situations though, an application can even meet its high-level goals by using appropriate monads within its core logic. Here are just a few applications that have monads at the heart of their designs: The Parsec parser library uses monads to combine simpler parsing rules into more complex ones, and is particularly useful for smaller domain-specific languages.

NET Framework that is heavily influenced by functional programming concepts, including core operators for composing queries monadically.

However, in those languages, "monad" is only shorthand for a function taking one parameter a function with two parameters being a "dyad", and so on. According to programming language researcher Philip Wadlercomputer scientist John C. Reynolds anticipated several facets of it in the 70s and early 80s, when he discussed the value of continuation-passing style, category theory as a rich source for formal semantics, and the type distinction between values and computations.

In earlier work, several computer scientists had advanced using category theory to provide semantics for the lambda calculus. Moggi's key insight was that a real-world program is not just a function from values to other values, but rather a transformation that forms computations on those values. When formalized in category-theoretic terms, this leads to the conclusion that monads are the structure to represent these computations. In particular, Haskell used a problematic "lazy stream" model up through v1.

Not only can the monad laws be used to check an instance's validity, but features from related structures like functors can be used through subtyping. Verifying the monad laws[ edit ] Returning to the Maybe example, its components were declared to make up a monad, but no proof was given that it satisfies the monad laws.

This can be rectified by plugging the specifics of Maybe into one side of the general laws, then algebraically building a chain of equalities to reach the other side: So to begin, a structure requires a higher-order function or "functional" named map to qualify as a functor: A monad's first transformation is actually the same unit from the Kleisli triple, but following the hierarchy of structures closely, it turns out unit characterizes an applicative functoran intermediate structure between a monad and a basic functor.

In the applicative context, unit is sometimes referred to as pure but is still the same function. What does differ in this construction is the law unit must satisfy; as bind is not defined, the constraint is given in terms of map instead: List [ edit ] The complex multivalued square and cube root functions may be composed so they produce the sixth root function.

The structure that governs the type of input and output and the structure that composes the different actions are, together, a list monad. Embedding a plain value in a list is also trivial in most languages: The difficulty with this approach is that bind expects monadic functions, which in this case will output lists themselves; as more functions are applied, layers of nested lists will accumulate, requiring more than a basic comprehension.

However, a procedure to apply any simple function over the whole list, in other words map, is straight-forward: But since Optional is a Functor it understands how to take doStuff and execute it on the value inside itself, then spit out a new Optional.

You don't lose the int-ness or optional-ness. Know what else can be a Functor? Applicatives Applicative lets you stuff the function itself inside a container, otherwise it's almost identical to Functors. I told you we were going to use plain english. Because you may want to have a function in the container and apply it to value s in another container of the same kind. You could extend Optional with apply like this: Swift doesn't have a built-in definition for applicative because you can't extend a protocol to say it returns the same container type but with a different element type.

Monad Monads are containers you can call flatMap on. Because a monad takes special care to avoid double-containering take that Oxford English Dictionary.