nLab subtype



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




In a dependent type theory with identity types, a subtype of a type AA is a type BB with a term i:BAi: B \subseteq A, with the inclusion relation defined as

BA[ f:BAis1Monic(f)]B \subseteq A \coloneqq \left[\sum_{f:B \to A} is1Monic(f)\right]

indicating the mere existence of a 1-monic function or embedding, where [T][T] is the propositional truncation of TT. AA is a supertype of BB.

This is different from a subtype with a chosen 1-monic function, a type BB with a term i:BAi:B \hookrightarrow A, with the type of 1-monic functions defined as

BA f:BAis1Monic(f)B \hookrightarrow A \coloneqq \sum_{f:B \to A} is1Monic(f)

in the same way that inhabited sets and pointed sets are different. However, equipping a type with an explicit 1-monic function is usually more useful in mathematics without the axiom of choice.

In a dependent type theory with identity types and a hierarchy of type universes a la Tarski, given a universe (𝒰,𝒯 𝒰)(\mathcal{U},\mathcal{T}_{\mathcal{U}}) a subtype of a type A:𝒰A:\mathcal{U} is a function B:𝒯 𝒰(A)Prop 𝒰B:\mathcal{T}_{\mathcal{U}}(A) \to \mathrm{Prop}_\mathcal{U}. In a universe these are the same as subtypes equipped with a 1-monic function as structure.

The type of all subtypes of BB in a universe is defined as

Sub 𝒰(B) A:𝒰𝒯 𝒰(A)BSub_\mathcal{U}(B) \coloneqq \sum_{A:\mathcal{U}} \mathcal{T}_\mathcal{U}(A) \subseteq B

See also

Last revised on June 17, 2022 at 17:42:33. See the history of this page for a list of all contributions to it.