nLab continuation monad

Idea

In a category with internal homs $[-,-]$, given an object $S$, the continuation monad is the endofunctor $X \mapsto [[X, S], S]$.

In computer science this monad (in computer science) is used to model continuation-passing style of programming, and therefore this is called the continuation monad. The idea here is that a morphism $f \colon X \to Y$ in the Kleisli category of the continuation monad, hence a morphism in the original category of the form $X\longrightarrow [[Y,S],S]$ is much like a map from $X$ to $Y$ only that instead of “returning” its output directly it instead feeds it into a given function $Y \to S$ which hence continues the computation.

Examples

• double negation monad

• dualizing object in a closed category

• extensive quantity

Related concepts

• selection monad

• continuation-passing style

• maybe monad

• state monad

References

As a monad in computer science:

• Tarmo Uustalu, p.27 of: Monads and Interaction Lecture 1 [pdf, pdf], lecture notes for MGS 2021 (2021):

The continuation monad is discussed in the generality of linear type theory as the linear double negation monad in

• Paul-André Melliès, Nicolas Tabareau, Linear continuation and duality, 2008 (pdf)

• Paul-André Melliès, The parametric continuation monad, Mathematical Structures in Computer Science, Festschrift in honor of Corrado Böhm for his 90th birthday (2013). (pdf)