nLab dependent type theory



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




Dependent type theory is the flavor of type theory that admits dependent types.

Its categorical semantics is in locally cartesian closed categories CC, where a dependent type

x:XE(x)type x : X \vdash E(x) \; \mathrm{type}

is interpreted as a morphism EXE \to X, hence an object in the slice category C /XC_{/X}.

Then change of context corresponds to base change in CC. See also dependent sum and dependent product.

Dependent type systems are heavily used for software certification.

In the foundations of mathematics

Dependent type theory itself support various foundations of mathematics via the propositions as some types interpretation of dependent type theory, where propositions are the types where every two elements are equal

isProp(A) x:A y:Ax= Ay\mathrm{isProp}(A) \coloneqq \prod_{x:A} \prod_{y:A} x =_A y

Suppose that a dependent type theory has a separate type judgment as well as dependent product types, dependent sum types, identity types, weak function extensionality, propositional truncations, empty type, unit type, sum types. All the operations in predicate logic are derivable from said type formers:



Judgments for types and terms

type theorycategory theory
typeobject in category
Atype\vdash\; A \; \mathrm{type}A𝒞A \in \mathcal{C}
a:A\vdash\; a \colon A*aA* \stackrel{a}{\to} A
dependent typeobject in slice category
x:XA(x)typex \colon X \;\vdash\; A(x) \; \mathrm{type}A X𝒞 /X\array{A \\ \downarrow \\ X} \in \mathcal{C}_{/X}
term in contextgeneralized elements/element in slice category
x:Xa(x):A(x)x \colon X \;\vdash \; a(x)\colon A(x)X a A id X X\array{X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{}} \\ && X}
x:Xa(x):Ax \colon X \;\vdash \; a(x)\colon AX (id X,a) X×A id X p 1 X\array{X &&\stackrel{(id_X,a)}{\to}&& X \times A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{p_1}} \\ && X}



The functors

constitute an equivalence of categories

DependentTypeTheoriesContLangLocallyCartesianClosedCategories. DependentTypeTheories \stackrel{\overset{Lang}{\leftarrow}}{\underset{Cont}{\to}} LocallyCartesianClosedCategories \,.

This (Seely, theorem 6.3). It is somewhat more complicated than this, because we need to strictify the category theory to match the category theory; see categorical model of dependent types. For a more detailed discussion see at relation between type theory and category theory.



For original references see at Martin-Löf dependent type theory, such as:

also published as:

Gentle exposition of the basic principles:

Introductory accounts:

Introduction with parallel details on using proof assistants:

for Coq:

for Agda:

Original discussion of dependent type theory as the internal language of locally cartesian closed categories is in

  • R. A. G. Seely, Locally cartesian closed categories and type theory, Math. Proc. Camb. Phil. Soc. (1984) 95 (pdf)

A formal definition of dependent type theories beyond Martin-Löf dependent type theory:

On (essentially algebraic) formulations of dependent type theory (see here at categorical models of dependent type theory):

For more see the references at Martin-Löf dependent type theory.

Last revised on January 7, 2024 at 14:47:06. See the history of this page for a list of all contributions to it.