nLab sequence extensionality


Set theory

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


Category theory



In set theory

Let \mathbb{N} be the set of natural numbers, and let AA be an arbitrary set. Then sequence extensionality states that for every sequence a:Aa:\mathbb{N} \to A and b:Ab:\mathbb{N} \to A, a= Aba =_{\mathbb{N} \to A} b if and only if a(n)= Ab(n)a(n) =_A b(n) for all natural numbers nn \in \mathbb{N}.

In category theory

Sequence extensionality in an arithmetic pretopos states that for every morphism a:Aa:\mathbb{N} \to A and b:Ab:\mathbb{N} \to A, a= Aba =_{\mathbb{N} \to A} b if and only if an= 𝟙Abna \circ n =_{\mathbb{1} \to A} b \circ n for all morphisms n:𝟙n:\mathbb{1} \to \mathbb{N}.

This definition makes sense in any finitely complete category with a parameterized natural numbers object.

In dependent type theory

In dependent type theory, sequence extensionality states that given two sequences a:Aa:\mathbb{N} \to A and b:Ab:\mathbb{N} \to A there is an equivalence of types between the identity type a= Aba =_{\mathbb{N} \to A} b and the dependent sequence type (n:)(a(n)= Ab(n))(n:\mathbb{N}) \to (a(n) =_{A} b(n)):

ΓAtypeΓ,a:A,b:Aseqext(a,b):(a= Ab)(n:)(a(n)= Ab(n))\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A, b:\mathbb{N} \to A \vdash \mathrm{seqext}(a, b):(a =_{\mathbb{N} \to A} b) \simeq (n:\mathbb{N}) \to (a(n) =_{A} b(n))}

There is also a version of sequence extensionality for dependent sequence types called dependent sequence extensionality, which states that given two dependent sequences a:(n:)A(n)a:(n:\mathbb{N}) \to A(n) and b:(n:)A(n)b:(n:\mathbb{N}) \to A(n) there is an equivalence of types between the identity type a= (n:)A(n)ba =_{(n:\mathbb{N}) \to A(n)} b and the dependent sequence type (n:)(a(n)= A(n)b(n))(n:\mathbb{N}) \to (a(n) =_{A(n)} b(n)):

Γ,n:A(n)typeΓ,a:(n:)A(n),b:(n:)A(n)dseqext(a,b):(a= (n:)A(n)b)(n:)(a(n)= A(n)b(n))\frac{\Gamma, n:\mathbb{N} \vdash A(n) \; \mathrm{type}}{\Gamma,a:(n:\mathbb{N}) \to A(n), b:(n:\mathbb{N}) \to A(n) \vdash \mathrm{dseqext}(a, b):(a =_{(n:\mathbb{N}) \to A(n)} b) \simeq (n:\mathbb{N}) \to (a(n) =_{A(n)} b(n))}


While function extensionality implies sequence extensionality, sequence extensionality is weaker than function extensionality. However, sequence extensionality is usually more relevant in dependent type theories which are strongly predicative, which usually have dependent sequence types but do not have general dependent function types.

 See also

Created on January 9, 2023 at 22:56:55. See the history of this page for a list of all contributions to it.