Editing Monad (category theory) (section)

====The state monad====

{{Expand section|date=February 2023|with=describe multiplication}}

Given a set <math>S</math>, the endofunctor of the '''state monad''' maps each set <math>X</math> to the set of functions <math>S\to S \times X</math>. The component of the unit at <math>X</math> maps each element <math>x\in X</math> to the function

:<math>\eta_X(x): S\to S\times X</math>
:<math>s\mapsto (s, x)</math>

The multiplication maps the function <math>f: S\to S\times (S\to S\times X), s\mapsto (s',f')</math> to the function

:<math>\mu_X(f): S\to S\times X</math>
:<math>s\mapsto f'(s')</math>

In functional programming and denotational semantics, the state monad models [[State (computer science)|stateful computations]].