nLab dependent function application to identifications


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
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity 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 set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
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




In dependent type theory, given types AA and BB and a family of elements x:Af(x):Bx:A \vdash f(x):B, the function application to identifications for f(x)f(x) is the family of elements

a:A,b:A,p:a= Abap f(a,b,p):f(a)= Bf(b)a:A, b:A, p:a =_A b \vdash \mathrm{ap}_f(a, b, p):f(a) =_B f(b)

A dependent function application to identifications is like an function application to identifications, but hwere we allow BB to depend on x:Ax:A and similarly the identity type f(a)= Bf(b)f(a) =_B f(b) to be a heterogeneous identity type and depend on the identification p:a= Abp:a =_A b.


In dependent type theory, given a type AA and a type family x:AB(x)x:A \vdash B(x) and a family of elements x:Af(x):B(x)x:A \vdash f(x):B(x), the dependent function application to identifications or dependent action on identifications for f(x)f(x) is the family of elements

a:A,b:A,p:a= Abapd f(a,b,p):hId B(a,b,p,f(a),f(b))a:A, b:A, p:a =_A b \vdash \mathrm{apd}_f(a, b, p):\mathrm{hId}_{B}(a, b, p, f(a), f(b))

inductively defined by

a:Aapd f(a,a,refl A(a))hrefl A(a,f(a)):hId B(a,a,refl A(a),f(a),f(a))a:A \vdash \mathrm{apd}_f(a, a, \mathrm{refl}_A(a)) \coloneqq \mathrm{hrefl}_A(a, f(a)):\mathrm{hId}_{B}(a, a, \mathrm{refl}_A(a), f(a), f(a))

where hId B(a,b,p,f(a),f(b))\mathrm{hId}_{B}(a, b, p, f(a), f(b)) is a heterogeneous identity type.

Using functions from the interval type

If dependent identification types are defined in terms of function types from the interval type; i.e.

p:𝕀A,x:B(p(0)),y:B(p(1))apd f(p):hId B(p,x,y)p:\mathbb{I} \to A, x:B(p(0)), y:B(p(1)) \vdash \mathrm{apd}_f(p):\mathrm{hId}_{B}(p, x, y)

then dependent function application is defined slightly differently.

By the recursion rule of the interval type, one could use a function p:𝕀Ap:\mathbb{I} \to A instead of elements a:Aa:A, b:Ab:A, and identification p:a= Abp:a =_A b. Then given a type AA and a type family x:AB(x)x:A \vdash B(x) and a family of elements x:Af(x):B(x)x:A \vdash f(x):B(x), the dependent function application to identifications or dependent action on identifications for f(x)f(x) is the family of elements

p:𝕀Aapd f(p):hId B(p,f(p(0)),f(p(1)))p:\mathbb{I} \to A \vdash \mathrm{apd}_f(p):\mathrm{hId}_{B}(p, f(p(0)), f(p(1)))

inductively defined using the induction principle of the path type 𝕀A\mathbb{I} \to A by

a:Aapd f(λi:𝕀.a)hrefl A(a,f(a)):hId B(λi:𝕀.a,f(a),f(a))a:A \vdash \mathrm{apd}_f(\lambda i:\mathbb{I}.a) \coloneqq \mathrm{hrefl}_A(a, f(a)):\mathrm{hId}_{B}(\lambda i:\mathbb{I}.a, f(a), f(a))

Using transport

In addition, there are two other families of elements which could be considered dependent function applications to identifications, which use transport and the inverse of transport rather than heterogeneous identity types:

a:A,b:A,p:a= Abapdl f(a,b,p):f(a)= B(a)tr B(a,b,p) 1(f(b))a:A, b:A, p:a =_A b \vdash \mathrm{apdl}_f(a, b, p):f(a) =_{B(a)} \mathrm{tr}_B(a, b, p)^{-1}(f(b))
a:A,b:A,p:a= Abapdr f(a,b,p):tr B(a,b,p)(f(a))= B(b)f(b)a:A, b:A, p:a =_A b \vdash \mathrm{apdr}_f(a, b, p):\mathrm{tr}_B(a, b, p)(f(a)) =_{B(b)} f(b)

Having one of the three notions of dependent function application to identifications means that one could define all three of them, because the types f(a)= B pf(b)f(a) =_B^p f(b), f(a)= B(a)tr B(a,b,p) 1(f(b))f(a) =_{B(a)} \mathrm{tr}_B(a, b, p)^{-1}(f(b)), and tr B(a,b,p)(f(a))= B(b)f(b)\mathrm{tr}_B(a, b, p)(f(a)) =_{B(b)} f(b) are all equivalent to each other.

See also

Last revised on December 26, 2023 at 20:19:33. See the history of this page for a list of all contributions to it.