nLab cubical path type

Contents

Context

Equality and Equivalence

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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

Cubical path types are a form of path types in dependent type theory used in cubical type theories.

A major difference between cubical path types and Martin-Löf identity types is the behaviour of the J rule. In Martin-Löf identity types the J rule holds up to definitional equality, but for cubical path types, the J rule only holds up to a path.

Another difference is that transport generally behaves better with cubical path types. Certain rules for the computation of transports in concrete type families hold up to definitional equality for cubical path types, but only up to an identification for Martin-Löf identity types.

Rules

The rules for (dependent) cubical path types are as follows

  • Formation
Γ,i:IAtypeΓa:[0/i]AΓb:[1/i]AΓpath i.A(a,b)type\frac{\Gamma, i:I \vdash A \; \mathrm{type} \quad \Gamma \vdash a:[0/i]A \quad \Gamma \vdash b:[1/i]A}{\Gamma \vdash \mathrm{path}_{i.A}(a,b) \; \mathrm{type}}
  • Introduction
Γ,i:Ia:AΓλi.a:path i.A([0/i]a,[1/i]a)\frac{\Gamma, i:I \vdash a:A}{\Gamma \vdash \lambda i.a:\mathrm{path}_{i.A}([0/i]a,[1/i]a)}
  • Elimination
Γp:path i.A(a,b)Γr:IΓp(r):[r/i]A[(r)[r0a|r1b]]\frac{\Gamma \vdash p:\mathrm{path}_{i.A}(a,b) \quad \Gamma \vdash r:I}{\Gamma \vdash p(r):[r/i]A [\partial(r) \to [r \equiv 0 \to a \vert r \equiv 1 \to b]]}
  • Computation
ΓctxΓ(λi.a)(r)[r/i]a:[r/i]A\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash (\lambda i.a)(r) \equiv [r/i]a:[r/i]A}
  • Uniqueness (optional)
ΓctxΓpλi.p(i):path i.A(a,b)\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash p \equiv \lambda i.p(i) : \mathrm{path}_{i.A}(a,b)}

In addition, there are coercion and composition operations which make the path type behave like an identity type:

  • Coercion:
Γ,i:IAtypeΓa:[r/i]AΓcoe i.A rra:[r/i]A[rra]\frac{\Gamma, i:I \vdash A \; \mathrm{type} \quad \Gamma \vdash a:[r/i]A}{\Gamma \vdash \mathrm{coe}^{r \rightsquigarrow r'}_{i.A} a:[r'/i]A \; [r' \equiv r \to a]}
  • Composition:

Regularity

In general, cubical path types in cubical type theory do not satisfy the J rule definitionally. However one could force the J rule to hold definitionally by appending a coercion regularity rule:

Γ,i:I,j:IA[j/i]AtypeΓcoe i.A rraa:[r/i]A\frac{\Gamma, i:I, j:I \vdash A \equiv [j/i]A \; \mathrm{type}}{\Gamma \vdash \mathrm{coe}^{r \rightsquigarrow r'}_{i.A} a \equiv a:[r'/i]A}

to the type theory.

See also

References

On why cubical path types and Martin-Löf identity types are different:

On cubical path types in XTT:

Last revised on September 30, 2022 at 01:31:38. See the history of this page for a list of all contributions to it.