Editing Monad (functional programming) (section)

===Environment monad===

{{See also|Monad (category theory)#The environment monad}}

An environment monad (also called a ''reader monad'' and a ''function monad'') allows a computation to depend on values from a shared environment. The monad type constructor maps a type {{mvar|T}} to functions of type {{math|''E'' → ''T''}}, where {{mvar|E}} is the type of the shared environment. The monad functions are:
<math display="block">\begin{array}{ll}
\text{return} \colon & T \rarr E \rarr T = t \mapsto e \mapsto t \\
\text{bind} \colon & (E \rarr T) \rarr (T \rarr E \rarr T') \rarr E \rarr T' = r \mapsto f \mapsto e \mapsto f \, (r \, e) \, e
\end{array}
</math>

The following monadic operations are useful:
<math display="block">\begin{array}{ll}
\text{ask} \colon & E \rarr E = \text{id}_E \\
\text{local} \colon & (E \rarr E) \rarr (E \rarr T) \rarr E \rarr T = f \mapsto c \mapsto e \mapsto c \, (f \, e)
\end{array}
</math>

The {{math|ask}} operation is used to retrieve the current context, while {{math|local}} executes a computation in a modified subcontext. As in a state monad, computations in the environment monad may be invoked by simply providing an environment value and applying it to an instance of the monad.

Formally, a value in an environment monad is equivalent to a function with an additional, anonymous argument; {{math|return}} and {{math|bind}} are equivalent to the {{math|K}} and {{math|S}} combinators, respectively, in the [[SKI combinator calculus]].