nLab
dependent type theory

Context

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

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive 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, (idemponent) 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

</table>

homotopy levels

semantics

Contents

Idea

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.

Description

Judgments for types and terms

type theorycategory theory
syntaxsemantics
judgmentdiagram
typeobject in category
A:Type\vdash\; A \colon TypeA𝒞A \in \mathcal{C}
termelement
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}

Properties

Theorem

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.

Examples

References

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

  • Agda Tutorial: Introduction to dependent type theory (webpage)

A 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)

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

Last revised on October 6, 2016 at 04:24:40. See the history of this page for a list of all contributions to it.