nLab polymonad


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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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




Unlike the bind operation of a monad which is of type

(α)(β)(mα(αmβ)mβ) (\forall\alpha) (\forall\beta) \big( m\alpha \to (\alpha\to m\beta) \to m\beta \big)

a polymonadic bind involves three abstract types:

(α)(β)(m 1α(αm 2β)m 3β). (\forall\alpha) (\forall\beta) \big( m_1\alpha\to(\alpha\to m_2\beta)\to m_3\beta \big) \,.

This notion subsumes monads in the sense that every set of monads and monad morphisms can be encoded using polymonads while obeying the polymonad laws. Moreover, it encompasses the monad-like programming patterns.

(Note that, from a category theoretic perspective, it is more natural to change the order of the arguments to match the form of Kleisli extension, so that the operations are of the form

(α)(β)((αm 2β)(m 1αm 3β)). (\forall\alpha) (\forall\beta) \big( (\alpha\to m_2\beta)\to (m_1\alpha\to m_3\beta) \big) \,.



A polymonad is the data of a collection \mathcal{M} of unary type constructors cointaining an element Id\mathrm{Id}, and a bind set Σ \Sigma_{\mathcal{M}} containing an element b Id,Id,Idb_{\mathrm{Id},\mathrm{Id},\mathrm{Id}} (reverse apply).

It must obey the following polymonad laws:

  • functorial law;
  • left identity;
  • right identity;
  • associativiy;
  • paired morphisms law;
  • composition closure law.

The first four laws are chosen to generalise monads, and the last two are chosen so that if certain binds can be uniquely defined in terms of other ones, then these must be present in Σ\Sigma.


  • Hoare monad?

  • effect monad?


Last revised on December 12, 2023 at 17:26:32. See the history of this page for a list of all contributions to it.