of all homotopy types
|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 homotopy type theory a type of (small) types is what semantically becomes a (small) object classifier. Thus, the type of types is a refinement of the type of propositions which only contains the (-1)-truncated/h-level-1 types (and is semantically a subobject classifier).
In the presence of a type of types a judgement of the form
\vdash A : Type
x : X \vdash A(x) : Type
says that is an -dependent type.
In homotopy type theory the type of types is often assumed to satisfy the univalence axiom. This is a reflection of the fact that in its categorical semantics as an object classifier is part of an internal (infinity,1)-category in the ambient (infinity,1)-topos: the one that as an indexed category is the small codomain fibration.
Per Martin-Lof’s original type theory contained a type of all types, which therefore in particular contained itself, i.e. one had . But it was pointed out by Jean-Yves Girard that this was inconsistent; see Girard's paradox. Thus, modern type theories generally contain a hierarchy of types of types, with and , etc.
Both Coq and Agda have systems to manage universe sizes and universe enlargement automatically; Agda’s is more advanced (universe polymorphism), whereas Coq’s is good enough for many purposes but tends to produce “universe inconsistencies” when working with univalence.
Detailed discussion of the type of types in Coq is in
See also around slide 8 of the survey