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 |
$\vdash\; A \colon Type$ | $A \in \mathcal{C}$ |
term | element |
$\vdash\; a \colon A$ | $* \stackrel{a}{\to} A$ |
dependent type | object in slice category |
$x \colon X \;\vdash\; A(x) \colon Type$ | $\array{A \\ \downarrow \\ X} \in \mathcal{C}_{/X}$ |
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.
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
For more see the references at Martin-Löf dependent type theory.