|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-level 1-type/h-prop|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator monad||modal type theory, monad (in computer science)|
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
x:A \; \vdash \;B(x):Type
is represented by a particular morphism , the intended meaning being that each type is the fiber of over . The morphism in a category may represent dependent types in this way are sometimes called display morphisms (especially when not every morphism is a display morphism).
When the theory of a category is phrased in dependent type theory, 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.