nLab quotient equivalence relation



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





Similar to the construction of a quotient set from a set (i.e. h-set) with an equivalence relation as a higher inductive type in homotopy type theory, there is a construction of a “quotient equivalence relation” from a type with two equivalence relations, as an higher inductive type family.

This construction is reminiscent of the quotient set construction in set theory, because equality in set theory is defined in the foundations as a equivalence relation. The difference is that here we treat both equivalence relations on the same level, while in set theory one of the equivalence relations is given primacy as the default equality of the foundations.


Recall that an equivalence relation on a type AA is a type family \sim consisting of dependent types aba \sim b indexed by objects a:Aa:A and b:Ab:A, with witnesses of reflexivity, symmetry, and transitivity, and (-1)-truncatedness

a:Arefl(a):(aa)a:A \vdash \mathrm{refl}(a):(a \sim a)
a:A,b:Asym(a,b):(ab)(ba)a:A, b:A \vdash \mathrm{sym}(a, b):(a \sim b) \to (b \sim a)
a:A,b:A,c:Atrans(a,b,c):(ab)×(bc)(ac)a:A, b:A, c:A \vdash \mathrm{trans}(a, b, c):(a \sim b) \times (b \sim c) \to (a \sim c)
a:A,b:Aproptrunc(a,b): p:ab q:abp= abqa:A, b:A \vdash \mathrm{proptrunc}(a, b):\prod_{p:a \sim b} \prod_{q: a \sim b} p =_{a \sim b} q

Given a type AA with two equivalence relations A\equiv_A and A\sim_{A}, the quotient equivalence relation on (A, A, A)(A, \equiv_{A}, \sim_{A}) is the higher inductive type family A \equiv^{'}_{A} inductively generated by the following constructors:

  • For each element a:Aa:A and b:Ab:A, a function

    f A(a,b):(a Ab)(a A b)f_{\equiv_A}(a, b):(a \equiv_{A} b) \to (a \equiv^{'}_{A} b)
  • For each element a:Aa:A and b:Ab:A, a function

    f A(a,b):(a Ab)(a A b)f_{\sim_A}(a, b):(a \sim_{A} b) \to (a \equiv^{'}_{A} b)
  • For each element a:Aa:A, a witness

    refl(a):(a A a)\mathrm{refl}(a):(a \equiv^{'}_{A} a)
  • For each element a:Aa:A and b:Ab:A, a witness

    sym(a,b,c):(a A b)(b A a)\mathrm{sym}(a, b, c):(a \equiv^{'}_{A} b) \to (b \equiv^{'}_{A} a)
  • For each element a:Aa:A, b:Ab:A, and c:Ac:A, a witness

    trans(a,b,c):(a A b)×(b A c)(a A c)\mathrm{trans}(a, b, c):(a \equiv^{'}_{A} b) \times (b \equiv^{'}_{A} c) \to (a \equiv^{'}_{A} c)
  • For each element a:Aa:A and b:Ab:A, a witness

    proptrunc(a,b): p:a A b q:a A bp= a A bq\mathrm{proptrunc}(a, b):\prod_{p:a \equiv^{'}_{A} b} \prod_{q: a \equiv^{'}_{A} b} p =_{a \equiv^{'}_{A} b} q

See also

Last revised on September 20, 2022 at 22:17:11. See the history of this page for a list of all contributions to it.