nLab sugaring



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




In computer science and to some extent also in formal logic, sugaring refers to the modification of formal notation (syntax) into a form which is more readable for humans, thereby “sweetening” it for consumption. Syntactic sugar does not add to the functionality or expressivity of the language.

Peter Landin first used the expression ‘syntactic sugaring’ in the context of interpretations of terms in the λ-calculus (Landin64).

The reverse process to expressions in the more formal language is sometimes known as ‘desugaring’.



In a set theory where classes are not “first-class objects” (for example ZFC, but not NBG), classes are only syntactic sugar and have no independent existence. For instance, if one defines a class M{X|φ(X)}M \coloneqq \{ X | \varphi(X) \} by comprehension, then “XMX \in M” is syntactic sugar for “φ(X)\varphi(X)”.


In formal linguistics, natural language versions of formal expressions are described as ‘sugarings’ (Ranta 1994). Often there are many ways to sugar a formal expression into natural language.

Consider the type theoretic expression x:man y:donkey(xownsy)\sum_{x: man} \sum_{y: donkey} (x\;owns\; y). Ranta (1994, p. 68) gives nine natural language sugarings for it by applying three different operations twice:

  • there is a man and there is a donkey and he owns it,
  • there is a man and he owns a donkey,
  • there is a man and there is a donkey that he owns,
  • there is a donkey and a man owns it,
  • a man owns a donkey,
  • there is a donkey that a man owns,
  • there is a man such that there is a donkey and he owns it,
  • there is a man who owns a donkey,
  • there is a man such that there is a donkey that he owns.


  • Peter Landin, The mechanical evaluation of expressions [pdf]

  • Aarne Ranta, Type theory and the informal language of mathematics, TYPES ‘93: Proceedings of the international workshop on Types for proofs and programs, pp. 352–365, (pdf)

  • Aarne Ranta, §1.6, §1.7, §9 in: Type-theoretical grammar, Oxford University Press (1994) [ISBN:9780198538578]

See also:

Last revised on October 14, 2023 at 08:52:57. See the history of this page for a list of all contributions to it.