natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
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
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.
They also seem to support a foundations of mathematics in terms of homotopy type theory.
type theory | category theory |
---|---|
syntax | semantics |
judgment | diagram |
type | object in category |
$\vdash\; A \; \mathrm{type}$ | $A \in \mathcal{C}$ |
term | element |
$\vdash\; a \colon A$ | $* \stackrel{a}{\to} A$ |
dependent type | object in slice category |
$x \colon X \;\vdash\; A(x) \; \mathrm{type}$ | $\array{A \\ \downarrow \\ X} \in \mathcal{C}_{/X}$ |
term in context | generalized elements/element in slice category |
$x \colon X \;\vdash \; a(x)\colon A(x)$ | $\array{X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{}} \\ && X}$ |
$x \colon X \;\vdash \; a(x)\colon A$ | $\array{X &&\stackrel{(id_X,a)}{\to}&& X \times A \\ & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{p_1}} \\ && X}$ |
The functors
$Cont$, that form a category of contexts of a dependent type theory;
$Lang$ that forms the internal language of a locally cartesian closed category
constitute an equivalence of categories
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:
Simon Thompson, §6.3 in: Type Theory and Functional Programming, Addison-Wesley (1991) [ISBN:0-201-41667-0, webpage, pdf]
Bart Jacobs, Chapter 10 in: Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, Elsevier (1998) [ISBN:978-0-444-50170-7, pdf]
(emphasis on the categorical model of dependent types)
Introduction with parallel details on using proof assistants:
for Coq:
Adam Chlipala, Certified programming with dependent types, MIT Press 2013 [ISBN:9780262026659, pdf, book webpage]
Théo Winterhalter, Formalisation and Meta-Theory of Type Theory, Nantes (2020) [pdf, github]
for Agda:
Ulf Norell, Dependently Typed Programming in Agda, p. 230-266 in: Advanced Functional Programming AFP 2008. Lecture Notes in Computer Science 5832 (2009) [doi:10.1007/978-3-642-04652-0_5, pdf]
Agda Tutorial: Introduction to dependent type theory (webpage)
Original discussion of dependent type theory as the internal language of locally cartesian closed categories is in
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 February 4, 2023 at 13:13:39. See the history of this page for a list of all contributions to it.