Monads

Origin. Godement (1958), Eilenberg-Moore (1965). Endofunctor with unit and multiplication. Encapsulates computational effects. Kleisli category for sequencing. Foundation for effectful programming.

Models. Monad T: C → C with η: Id → T, μ: T² → T. Laws: associativity and identity. Algebras: structured objects. Computational interpretation: effects.

Formalism.

Monad: (T, η, μ) where T: C → C η: Id_C → T (unit) μ: T ∘ T → T (multiplication)

Monad laws: μ ∘ Tμ = μ ∘ μT (associativity) μ ∘ Tη = id_T = μ ∘ ηT (unit laws)

Kleisli category: Objects: same as C. Morphisms A → B: morphisms A → TB in C. Composition: g ∘_K f = μ ∘ Tg ∘ f

Kleisli triple: (T, η, —) where —: (A → TB) → (TA → TB) f*: TA → TB extends f: A → TB

Examples: Maybe/Option: T(A) = A + 1 (partial functions) List: T(A) = A* (nondeterminism) State: T(A) = S → (A × S) (stateful computation) Continuation: T(A) = (A → R) → R

Eilenberg-Moore category: Objects: T-algebras (A, α: TA → A) Structure maps satisfying laws.

Commutative monad: Tμ ∘ τ = μ ∘ Tτ Enables parallel composition.

Symbols.

Symbol	Unicode	Name	Meaning
T	—	Monad	Endofunctor
η	U+03B7	Unit	Return
μ	U+03BC	Multiplication	Join
—*	—	Kleisli extension	Bind
>>=	—	Bind	Sequencing

Metatheory. Free-forgetful adjunction induces monad. Every monad from adjunction. Distributive laws for composition. Monad transformers.

Applies to. Functional programming (Haskell). Effect systems. Denotational semantics. Database queries. Probabilistic programming.

Limitations. Composition not automatic. Transformer overhead. Not all effects monadic. Learning curve.