Contents

Contents

Idea

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

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

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

is interpreted as a morphism $E \to X$, hence an object in the slice category $C_{/X}$.

Then change of context corresponds to base change in $C$. 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

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

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:

Then

Description

Judgments for types and terms

type theorycategory theory
syntaxsemantics
judgmentdiagram
typeobject in category
$\vdash\; A \; \mathrm{type}$$A \in \mathcal{C}$
termelement
$\vdash\; a \colon A$$* \stackrel{a}{\to} A$
dependent typeobject in slice category
$x \colon X \;\vdash\; A(x) \; \mathrm{type}$$\array{A \\ \downarrow \\ X} \in \mathcal{C}_{/X}$
term in contextgeneralized elements/element in slice category
$x \colon X \;\vdash \; a(x)\colon A(x)$$\array{X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{}} \\ && X}$
$x \colon X \;\vdash \; a(x)\colon A$$\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

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

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.

References

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 August 20, 2024 at 17:09:45. See the history of this page for a list of all contributions to it.