nLab
interval type

Context

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 rulecompositionsubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator 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

homotopy levels

semantics

Contents

Idea

The interval type is an axiomatization of the cellular interval object in the context of homotopy type theory.

Definition

As a higher inductive type, the interval is given by

Inductive Interval : Type
  | left : Interval
  | right : Interval
  | segment : Id Interval left right

This says that the type is inductive constructed from two terms in the interval, whose interpretation is as the endpoints of the interval, together with a term in the identity type of paths between these two terms, which interprets as the 1-cell of the interval

leftsegmentright. left \stackrel{segment}{\to} right \,.

Properties

  • An interval type is provably contractible. Conversely, any contractible type satisfies the rules of an interval type up to propositional equality.

  • On the other hand, postulating an interval type with definitional computation rules for left and right implies function extensionality. (Shulman).

References

Revised on December 10, 2011 00:48:04 by Mike Shulman (71.136.231.40)