Monad

Summary

A monad on a category is an endofunctor $T$ with a unit $\eta \colon \mathrm{Id} \Rightarrow T$ and a multiplication $\mu \colon T T \Rightarrow T$ that satisfy associativity and unit laws. It is the categorical form of a monoid, with composition of functors in place of multiplication. Every pair of adjoint functors gives a monad, and every monad comes from an adjunction. Monads describe algebraic structure: groups, monoids, rings and many other algebraic categories are the algebras of a monad on sets.