nLab 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




The type-theoretic version of the fact that in set theory functions preserve equality between elements in sets.


Basic definition

In dependent type theory, given types AA and BB and a function x:Af(x):Bx \colon A \vdash f(x) \colon B, there is induced a dependent function between the corresponding identity types

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

defined by Id-induction from

ap f(refl a)refl f(a) ap_f\big(refl_a\big) \,\equiv\, refl_{f(a)}

and called the application or action of ff to/on identities/identifications/equalities/paths [e.g. UFP (2013, p. 69), Rijke (2022, §5.3)].

By repeated Id-induction it readily follows, stagewise (e.g. UFP13, Lem 2.2.1), that ap fap_f respects the concatenation, and the inversion of identifications, up to coherent higher-order identifications, hence that it acts as an \infty -functor on the \infty -groupoid-structure on homotopy types :

As the non-dependent version of the dependent function application to identifications

The function application to identifications is a special case of the dependent function application to identifications for which the type family x:ABx:A \vdash B is a constant type family, and thus the dependent identity type f(a)= B pf(b)f(a) =_B^p f(b) doesn’t depend on the path p:a= Abp:a =_A b and is thus a normal identity type f(a)= Bf(b)f(a) =_B f(b).

Inductive definition

If the function application to identifications is inductively defined, then it comes with rules saying that the following judgment can be formed

a:Aap f(a,a)(refl A(a))refl B(f(a)):Ω(B,f(a))a:A \vdash \mathrm{ap}_{f}(a, a)(\mathrm{refl}_{A}(a)) \equiv \mathrm{refl}_{B}(f(a)):\Omega(B, f(a))

where Ω(A,a)\Omega(A, a) is the loop space type a= Aaa =_A a of AA at a:Aa:A.

Binary function applications to identifications

Given types AA, BB, and CC and a binary function x:A,y:Bf(x,y):Cx:A, y:B \vdash f(x, y):C, there is a dependent function

a:A,a:A,b:B,b:B,p:a= Aa,q:b= Bbapbinary f(a,a,b,b)(p,q):f(a,b)= Cf(a,b)a:A, a':A, b:B, b':B, p:a =_A a', q:b =_B b' \vdash \mathrm{apbinary}_f(a, a', b, b')(p, q):f(a, b) =_C f(a', b')

called the binary function application to identifications or binary action on identifications (Rijke (2022), §19.5), inductively defined by

a:A,b:Bapbinary f(a,a,b,b)(refl A(a),refl B(b))refl C(f(a,b)):Ω(C,f(a,b))a:A, b:B \vdash \mathrm{apbinary}_{f}(a, a, b, b)(\mathrm{refl}_{A}(a), \mathrm{refl}_{B}(b)) \equiv \mathrm{refl}_{C}(f(a, b)):\Omega(C, f(a, b))

where Ω(A,a)\Omega(A, a) is the loop space type a= Aaa =_A a of AA at a:Aa:A.

Binary function applications to identifications are used in proving product extensionality for product types, as well as defining multiplication on the integers apbinary μ:×\mathrm{apbinary}_{\mu}:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} from multiplication on the circle type μ:S 1×S 1S 1\mu:S^1 \times S^1 \to S^1 when the integers are defined as the loop space type of the element base:S 1\mathrm{base}:S^1, Ω(S 1,base)\mathbb{Z} \coloneqq \Omega(S^1, \mathrm{base}).

Finitary function application to identifications

Let nn be a natural number, let AA be a family of types indexed by the finite type with nn elements Fin(n)\mathrm{Fin}(n), and let BB be a type. Then given a function

n:,x:( i:Fin(n)A(i))f(x):Bn:\mathbb{N}, x:\left(\prod_{i:\mathrm{Fin}(n)} A(i)\right) \vdash f(x):B

there is a dependent function

n:,a: i:Fin(n)A(i),b: i:Fin(n)A(i)ap f(n)(a,b):( i:Fin(n)a(i)= A(i)b(i))f(a)= Bf(b)n:\mathbb{N}, a:\prod_{i:\mathrm{Fin}(n)} A(i), b:\prod_{i:\mathrm{Fin}(n)} A(i) \vdash \mathrm{ap}_f(n)(a, b):\left(\prod_{i:\mathrm{Fin}(n)} a(i) =_{A(i)} b(i)\right) \to f(a) =_{B} f(b)

See also


Last revised on January 23, 2023 at 19:31:21. See the history of this page for a list of all contributions to it.