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
In type theory, a dependent type or type in context is a family or bundle of types which vary over the elements (terms) of some other type. It can be regarded as a formalization of the notion of “indexed family,” providing a structural account of families (in contrast to the material approach which requires sets to be able to contain other sets as elements).
Type theory with the notion of dependent types is called dependent type theory.
In the categorical semantics of type theory, a dependent type
is represented by a particular morphism , the intended meaning being that each type is the fiber of over . The morphism in a category that may represent dependent types in this way are sometimes called display morphisms (especially when not every morphism is a display morphism).
Dependent types can be thought of as fibrations in classical homotopy theory. The base space is , the total space is and the fiber . This gives the fibration:
When the theory of a category is phrased in dependent type theory then there is one type “” of objects and a type of morphisms, which is dependent on two terms of type , so that for any there is a type of arrows from to . This dependency is usually written as . In some theories, it makes sense to say that the type of “” itself is (usually understood as or ), i.e. a function from pairs of elements of to the universe of types.
In Coq:
Last revised on June 9, 2022 at 00:31:04. See the history of this page for a list of all contributions to it.