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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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) : 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.

They also seem to support a foundations of mathematics in terms of homotopy type theory.


Judgments for types and terms

type theorycategory theory
typeobject in category
A:Type\vdash\; A \colon TypeA𝒞A \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) \colon TypeA 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.



All the essential ingredients are listed in

In part I there the standard type formation, term introduction/term elimination and computation rules of dependent type theory are listed.

An introduction with parallel details on Coq-programming is in

See also

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

A formal definition of dependent type theories is given in

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