nLab dependent identity 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

In dependent type theory, given a type AA, a type family x:AB(x)x:A \vdash B(x), terms a 0:Aa_0:A, a 1:Aa_1:A, and an identification p:a 0= Aa 1p:a_0 =_A a_1, a dependent identity type or a heterogeneous identity type between two elements b 0:B(a 0)b_0: B(a_0) and b 1:B(a 1)b_1:B(a_1) is a type whose elements witness that b 0b_0 and b 1b_1 are “equal” over or modulo the identification pp. There are different ways to define this precisely, depending partly on the particular type theory used.

Definition in Martin-Löf type theory

One way to define the dependent identity type in Martin-Lof type theory is using transport along the identification pp:

(a= B pb)(tr B p(a)= B(b)b)(a =_B^p b) \coloneqq (\mathrm{tr}_B^p(a) =_{B(b)} b)

There are also other possibilities…

Definition in higher observational type theory

In higher observational type theory, the dependent identity type is a primitive type former (although depending on the presentation, it can also be obtained using apap into the universe). In its general form, the type family can depend not just on a single type but on a type telescope Δ\Delta. The resulting dependent identity type then depends on an “identification in that telescope”, which is defined by mutual recursion as a telescope of dependent identity types. The formation rule is then

ς:δ= Δδ δAtypea:A[δ]a :A[δ ]a= Δ.A ςa type\frac{\varsigma:\delta =_\Delta \delta^{'} \quad \delta \vdash A\; \mathrm{type} \quad a:A[\delta] \quad a^{'}:A[\delta^{'}]}{a =_{\Delta.A}^\varsigma a^{'}\; \mathrm{type}}

… needs to be finished

Dependent identity types in universes

Given a term of a universe A:𝒰A:\mathcal{U}, a judgment z:𝒯 𝒰(A)B:𝒰z:\mathcal{T}_\mathcal{U}(A) \vdash B:\mathcal{U}, terms x:𝒯 𝒰(A)x:\mathcal{T}_\mathcal{U}(A) and y:𝒯 𝒰(A)y:\mathcal{T}_\mathcal{U}(A), and an identity p:id 𝒯 𝒰(A)(x,y)p:\mathrm{id}_{\mathcal{T}_\mathcal{U}(A)}(x,y), we have

ap z.B(p):id 𝒰(B(x),B(y))\mathrm{ap}_{z.B}(p):\mathrm{id}_\mathcal{U}(B(x),B(y))

and

(u,v):𝒯 𝒰(B(x))×𝒯 𝒰(B(y))π 1((ap z.B(p)))(u,v):𝒰(u,v):\mathcal{T}_\mathcal{U}(B(x)) \times \mathcal{T}_\mathcal{U}(B(y)) \vdash \pi_1(\nabla(\mathrm{ap}_{z.B}(p)))(u,v):\mathcal{U}

We could define a dependent identity type as

id 𝒯 𝒰(z.B) p(u,v)π 1((ap z.B(p)))(u,v)\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{p}(u, v) \coloneqq \pi_1(\nabla(\mathrm{ap}_{z.B}(p)))(u, v)

There is a rule

A:𝒰z:𝒯 𝒰(A)B:𝒰a:𝒯 𝒰(A)id 𝒯 𝒰(z.B) refl a(u,v)id 𝒯 𝒰(B[a/z])(u,v)\frac{A:\mathcal{U} \quad z:\mathcal{T}_\mathcal{U}(A) \vdash B:\mathcal{U} \quad a:\mathcal{T}_\mathcal{U}(A)}{\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{\refl_{a}}(u, v) \equiv \mathrm{id}_{\mathcal{T}_\mathcal{U}(B[a/z])}(u, v)}

and for constant families B:𝒰B:\mathcal{U}

id 𝒯 𝒰(z.B) p(u,v)id 𝒯 𝒰(B)(u,v)\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{p}(u, v) \equiv \mathrm{id}_{\mathcal{T}_\mathcal{U}(B)}(u, v)

Categorical semantics

needs to be written

See also

type, type theory

dependent type, dependent type theory, Martin-Löf dependent type theory

homotopy type, homotopy type theory

References

Last revised on June 6, 2022 at 11:45:49. See the history of this page for a list of all contributions to it.