Monad

In category theory, a monad on a category C is a triple (T, η, μ) in which T is an endofunctor T: C → C and η: Id_C → T and μ: T ∘ T → T are natural transformations. They satisfy the associativity and unit laws: μ ∘ Tμ = μ ∘ μT and μ ∘ Tη = μ ∘ ηT = id_T.

In functional programming, a monad is a design pattern that represents computations wrapped in a context, such as optional values, failure, state, or I/O. In languages like Haskell, a monad is a type constructor M with operations return :: a -> M a and (>>=) :: M a -> (a -> M b) -> M b, and often fmap for Functor. The monad laws—left identity, right identity, and associativity—guarantee that chaining computations is predictable and composable.

Common examples include Maybe/Option for computations that may fail, List for nondeterministic computations, IO for input/output actions, State for threaded state, and Reader for read-only environments. Monads provide a uniform interface to sequence computations with effects while preserving purity, enabling modular construction of complex behaviors. Monads are frequently combined with monad transformers to stack multiple effects, and with syntactic sugar like do-notation or comprehensions to streamline sequencing.