nLab uniqueness quantifier

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

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
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

semantics

Contents

Definition

In first order logic with equality

Uniqueness up to equality

In first order logic with equality, given a predicate PP on a type TT with equality, the uniqueness quantifier of PP, denoted !x:T,P(x)\exists!\, x\colon T, P(x), is defined in terms of the universal and existential quantifiers as

!x:T,P(x)x:T,P(x)y:T,P(y)(x=y). \exists!\, x\colon T, P(x) \;\equiv\; \exists\, x\colon T, P(x) \wedge \forall\, y\colon T, P(y) \Rightarrow (x = y) .

The intended interpretation is that !x:T,P(x)\exists!\, x\colon T, P(x) is true iff P(a)P(a) is true for exactly one element aa of TT.

Uniqueness up to isomorphism

Sometimes, we want to use a weaker notion of equivalence than strict equality, such as isomorphism xyx \cong y. The uniqueness up to isomorphism quantifier of PP, denoted ! x:T,P(x)\exists!_\cong\, x\colon T, P(x), is defined in terms of the universal and existential quantifiers as

! x:T,P(x)x:T,P(x)y:T,P(y)(xy). \exists!_\cong\, x\colon T, P(x) \;\equiv\; \exists\, x\colon T, P(x) \wedge \forall\, y\colon T, P(y) \Rightarrow (x \cong y).

The intended interpretation is that ! x:T,P(x)\exists!_\cong\, x\colon T, P(x) is true iff P(a)P(a) is true for exactly one element aa of TT up to isomorphism.

Uniqueness up to isomorphism quantifiers are important in category theory, where the relevant notion of sameness is isomorphism rather than strict equality. It is also important in foundational set theories where the type of sets does not have equality, such as some presentations of SEAR and ETCS.

 In dependent type theory

In dependent type theory, given a type TT and a type family x:TP(x)x:T \vdash P(x), the uniqueness quantifier is a type defined as

!x:T.P(x)isContr( x:TP(x))\exists!\, x\colon T. P(x) \coloneqq \mathrm{isContr}\left(\sum_{x:T} P(x)\right)

which indicates that the dependent sum type x:TP(x)\sum_{x:T} P(x) is a contractible type, which is only the case for a family of type if every dependent type is a mere proposition and, for exactly one element x:Tx:T up to identity, the type P(x)P(x) is inhabited.

Usages

Defining exclusive disjunction

In dependent type theory, given two mere propositions PP and QQ, by descent or large elimination? of the type of booleans, one can construct a boolean-indexed family of propositions

x:boolrec bool P,Q(x)x:\mathrm{bool} \vdash \mathrm{rec}_\mathrm{bool}^{P, Q}(x)

with equivalences of types

β bool P,Q(0):rec bool P,Q(0)Pβ bool P,Q(1):rec bool P,Q(1)Q\beta_\mathrm{bool}^{P, Q}(0):\mathrm{rec}_\mathrm{bool}^{P, Q}(0) \simeq P \quad \beta_\mathrm{bool}^{P, Q}(1):\mathrm{rec}_\mathrm{bool}^{P, Q}(1) \simeq Q

in the case for descent for booleans, or with judgmental equality of types

rec bool P,Q(0)Prec bool P,Q(1)P\mathrm{rec}_\mathrm{bool}^{P, Q}(0) \equiv P \quad \mathrm{rec}_\mathrm{bool}^{P, Q}(1) \equiv P

in the case for large elimination for booleans.

The uniqueness quantifier of the above family of propositions is the exclusive disjunction of PP and QQ:

P̲Q!x:𝟚.rec bool P,Q(x)P \underline{\vee} Q \coloneqq \exists!x:\mathbb{2}.\mathrm{rec}_\mathrm{bool}^{P, Q}(x)

Bijections and equivalences

The uniqueness quantifier is used in the definition of a bijection in set theory and an equivalence in type theory, where one defines a function f:ABf:A \to B to be a bijection or equivalence if for all y:By:B the there is a unique x:Ax:A such that f(x)= Byf(x) =_B y

isEquiv(f)y:B.!x:A.f(x)= By\mathrm{isEquiv}(f) \coloneqq \forall y:B.\exists! x:A.f(x) =_B y

In dependent type theory, this is the same as defining a family of elements x:Af(x):Bx:A \vdash f(x):B to be an equivalence if it comes with a family of elements

y:Bϵ(f)(y):!x:A.f(x)= Byy:B \vdash \epsilon(f)(y):\exists! x:A.f(x) =_B y

The inverse of an equivalence is given by the family of elements y:Bϵ A(ϵ(f)(y)):Ay:B \vdash \epsilon_A(\epsilon(f)(y)):A, where ϵ A\epsilon_A is defined in the elimination rules for uniqueness quantifiers in dependent type theory.

Anafunctions

Similarly, uniqueness quantifications is used in the definition of an anafunction, where one defines a relation or correspondence x:A,y:BR(x,y)x:A, y:B \vdash R(x, y) to be an anafunction if for all x:Ax:A there is a unique y:By:B such that R(x,y)R(x, y)

isAnafunc(R)x:A.!y:B.R(x,y)\mathrm{isAnafunc}(R) \coloneqq \forall x:A.\exists! y:B.R(x, y)

Univalent universes

Uniqueness quantifiers are also used to define univalent universes. A Russell universe UU is a univalent universe if for all elements A:UA:U there is a unique B:UB:U such that ABA \simeq B:

isUnivalent(U)A:U.!B:U.AB\mathrm{isUnivalent}(U) \coloneqq \forall A:U.\exists! B:U.A \simeq B

Similarly, a Tarski universe (U,T)(U, T) is a univalent universe if for all elements A:UA:U there is a unique B:UB:U such that T(A)T(B)T(A) \simeq T(B):

isUnivalent(U,T)A:U.!B:U.T(A)T(B)\mathrm{isUnivalent}(U, T) \coloneqq \forall A:U.\exists! B:U.T(A) \simeq T(B)

References

Last revised on May 14, 2025 at 23:22:07. See the history of this page for a list of all contributions to it.