nLab continuation monad



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




In a category with internal homs [,][-,-], given an object SS, the continuation monad is the endofunctor X[[X,S],S]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:XYf \colon X \to Y in the Kleisli category of the continuation monad, hence a morphism in the original category of the form X[[Y,S],S]X\longrightarrow [[Y,S],S] is much like a map from XX to YY only that instead of “returning” its output directly it instead feeds it into a given function YSY \to S which hence continues the computation.



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)

Last revised on December 30, 2020 at 05:27:19. See the history of this page for a list of all contributions to it.