nLab monad transformer



Categorical algebra

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 computer science and specifically in type theory, a monad transformer is a type constructor which takes a monad as an argument and returns a monad as a result. The concept is typically treated in the literature on monads in computer science.

Monad transformers generally derive from ordinary monads and allow a modular composition, so that the action on the identity monad of the associated transform MTM T of a monad MM is equivalent to MM.

This construction is sometimes viewed (see HP07, Eff) as a complication resulting from passing to monads from the setting of Lawvere theories, where any two theories may be naturally combined.


Textbook account:

See also

  • Bryan O’Sullivan, Don Stewart, and John Goerzen, Monad transformers, Chapter 18 of Real World Haskell.

  • Chung-Chieh Shan, Monad transformers, blog post

  • Mauro Jaskelioff, Eugenio Moggi, Monad Transformers as Monoid Transformers, pdf

  • Oleksandr Manzyuk, Calculating monad transformers with category theory, pdf

  • Martin Hyland, John Power, The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads, Electronic Notes in Theoretical Computer Science (ENTCS) archive Volume 172, April, 2007 Pages 437-458 (pdf)

Last revised on May 19, 2021 at 07:36:44. See the history of this page for a list of all contributions to it.