nLab term



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 formal logic such as type theory a term zz is an entity/expression of the formal language which is of some type ZZ. One writes z:Zz : Z to express this (a typing judgement). The semantics of terms in Set is: elements of a set, where one writes zZz \in Z. One also says zz is an inhabitant of the type ZZ and that ZZ is an inhabited type if it has a term.

A term z:Zz : Z may depend on free variables xx that are themselves terms x:Xx : X of some other type XX. For instance zx+3z \coloneqq x + 3 may be a term of type ZZ \coloneqq \mathbb{Z} (the integers) which depends on a variable term xx also of type XX \coloneqq \mathbb{Z} the integers. The notation for this in the metalanguage is

x:Xz:Z. x : X \vdash z : Z \,.

Generally here also the type ZZ itself may depend on the variable xx (hence the term zz may be of different type depending on its free variables) in which case one says that zz is a term of XX-dependent type.


In the metalanguage of type theory called natural deduction, terms are what the term introduction rules produce.

Categorical semantics

Here are comments on the interpretation of types in the categorical semantics of dependent type theory.

In the internal language of any category CC, a morphism

f:BA f : B \to A

is a term f(x)f(x) of type AA where xx is a free variable of type BB, which in type-theoretic symbols is given by

x:Bf(x):A. x\colon B \vdash f(x)\colon A \,.

Last revised on May 7, 2022 at 12:46:31. See the history of this page for a list of all contributions to it.