nLab cone type

Contents

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 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

semantics

Contents

Idea

The cone type is an axiomatization of the cone in the context of homotopy type theory.

Definition

A cone type on a type AA is a higher inductive type Cone(A)Cone(A) inductively generated by

  • A term e:Cone(A)e:Cone(A)

  • A function i:ACone(A)i: A \to Cone(A)

  • A term

p: a:Ae=i(a)p: \prod_{a:A} e = i(a)

In Coq pseudocode, the cone is given by

Inductive Cone (A : Type) : Type
  | vertex : Cone A
  | base :  A -> Cone A
  | edge : A -> Id Cone A vertex base(a)

It can equivalently be defined as the (homotopy) pushout of AA and the unit type 11 under AA. This definition makes it clear that the cone type is always a contractible type. As a result, cone types could be though of as a way of constructing free contractible types for any type AA.

Examples

  • The unit type 11 is the cone type of an empty type 00.

  • The interval type II is the cone type of the unit type 11

  • A simplex type? Δ n\Delta_n is the cone type of the simplex type Δ n1\Delta_{n-1}. Δ 1=I\Delta_1 = I and Δ 0=1\Delta_0 = 1.

See also

Last revised on June 16, 2022 at 07:58:14. See the history of this page for a list of all contributions to it.