Editing Monad (category theory) (section)

===Monads used in denotational semantics===

{{See also|Monad (functional programming)}}

The following monads over the category of sets are used in [[denotational semantics]] of [[Imperative programming language|imperative programming languages]], and analogous constructions are used in functional programming.

====The maybe monad====

The endofunctor of the '''maybe''' or '''partiality''' monad adds a disjoint point:{{sfn|Riehl|2017|p=155}}

:<math>(-)_{*}: \mathbf{Set}\to\mathbf{Set}</math>
:<math>X\mapsto X\cup\{*\}</math>

The unit is given by the inclusion of a set <math>X</math> into <math>X_*</math>:

:<math>\eta_X: X\to X_{*}</math>
:<math>x\mapsto x</math>

The multiplication maps elements of <math>X</math> to themselves, and the two disjoint points in <math>(X_{*})_{*}</math> to the one in <math>X_{*}</math>.

In both functional programming and denotational semantics, the maybe monad models [[Partial function|partial computations]], that is, computations that may fail.

====The state monad====

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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]].

====The environment monad====

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Given a set <math>E</math>, the endofunctor of the '''reader''' or '''environment monad''' maps each set <math>X</math> to the set of functions <math>E\to X</math>. Thus, the endofunctor of this monad is exactly the [[hom functor]] <math>\mathrm{Hom}(E, -)</math>. The component of the unit at <math>X</math> maps each element <math>x\in X</math> to the [[constant function]] <math>e\mapsto x</math>.

In functional programming and denotational semantics, the environment monad models computations with access to some read-only data.

====The list and set monads====

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The '''list''' or '''nondeterminism monad''' maps a set ''X'' to the set of finite [[Sequence|sequences]] (i.e., [[List (abstract data type)|lists]]) with elements from ''X''. The unit maps an element ''x'' in ''X'' to the singleton list [x]. The multiplication concatenates a list of lists into a single list.

In functional programming, the list monad is used to model [[Nondeterministic model of computation|nondeterministic computations]]. The covariant powerset monad is also known as the '''set monad''', and is also used to model nondeterministic computation.