nLab typal equality


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


Equality and Equivalence



The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms



Just as there are two notions of judgmental equality in dependent type theory, judgmental equality of terms and judgmental equality of types, there are two types in dependent type theory which could be considered typal equality:

In type universes UU, this means that small types A:UA:U have two notions of typal equality, one from the identity type between small types, and the other from the equivalence type between small types, and the univalence axiom states that these two types are equivalent (i.e. typally equal) to each other.

 Parallels between judgmental equality and typal equality

The parallels between the structural rules for judgmental equality and typal equality are shown below:

judgmental equalitytypal equality
judgmental equality of termsidentification
reflexivity of judgmental equality of termsidentity identification
symmetry of judgmental equality of termsinverse identification
transitivity of judgmental equality of termscomposition of identifications
judgmental equality of typesequivalence
reflexivity of judgmental equality of typesidentity equivalence
symmetry of judgmental equality of typesinverse equivalence
transitivity of judgmental equality of typescomposition of equivalences
principle of substitutiontransport
variable conversion rulesubstitution of evaluation of inverse equivalence

Homogeneous and heterogeneous typal equality

homogeneous identificationheterogeneous identificationhomogeneous equivalenceheterogeneous equivalence
typea= Aba =_A bx= B pyx =_B^p yABA \simeq BABA \simeq B
identity termrefl A(a):a= Aa\mathrm{refl}_A(a):a =_A aapd B p(f):f(a)= B pf(b)\mathrm{apd}_B^p(f):f(a) =_B^p f(b)id A:AA\mathrm{id}_A:A \simeq Atr B p:B(a)B(b)\mathrm{tr}_B^p:B(a) \simeq B(b)

 See also

Last revised on January 19, 2023 at 23:13:27. See the history of this page for a list of all contributions to it.