nLab denotational semantics

Denotational semantics



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


Denotational semantics


In computer science and formal logic, denotational semantics refers semantics based on the idea that programs and the data they manipulate are symbolic realizations of abstract mathematical objects.

For example, the denotational semantics

The idea of denotational semantics is thus to associate an appropriate mathematical object, such as a number, a tuple, or a function, with each term of the given programming language.

A key requirement on denotational semantics is that it respects the compositionality of programming languages, hence that the semantics of terms constructed from sub-terms is correspondingly built from the semantics of these sub-terms.

Under the Curry-Howard correspondence, also proofs can be seen as programs and thus one can apply denotational semantics to proofs. In this way proofs are interpreted as functions or more generally as morphisms of some category. This is used to interpret proofs of intuitionistic logic and linear logic as morphisms of respectively cartesian closed categories and star-autonomous categories, as discussed at relation between category theory and type theory. For more general such semantics see at categorical semantics of dependent type theory.


Denotational semantics originates with the proposal of domain theory to regard data types as posets ((0,1)-categories):

  • Dana S. Scott, Outline of a mathematical theory of computation, in: Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems (1970) 169–176. [pdf, pdf]

  • Dana S. Scott, Christopher Strachey, Toward a Mathematical Semantics for Computer Languages, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [pdf, pdf]

  • Dana Scott, Data types as lattices. SIAM Journal of Computing 5 3 (1976) 522–587 [doi:10.1137/0205037, pdf]

Lectures and introductions:

Textbook accounts:

See also:

Discussion of denotational semantics for Haskell:

Last revised on March 5, 2023 at 06:15:34. See the history of this page for a list of all contributions to it.