nLab definitional isomorphism

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

Equality and Equivalence

Contents

Definition

In dependent type theory, a definitional isomorphism or defintional section-retraction or judgmental isomorphism between two types AA and BB consists of functions f:ABf:A \to B and f 1:BAf^{-1}:B \to A such that given any term x:Ax:A, the term f 1(f(x))f^{-1}(f(x)) is judgmentally equal to xx and given any term y:By:B, the term f(f 1(y))f(f^{-1}(y)) is definitionally equal to yy. In symbols,

x:Af 1(f(x))x:Aandy:Bf(f 1(y))y:Bx:A \vdash f^{-1}(f(x)) \equiv x:A \quad \mathrm{and} \quad y:B \vdash f(f^{-1}(y)) \equiv y:B

If the dependent type theory has identity types, definitional isomorphisms are, in particular, equivalences since the homotopies and coherence law for (half-adjoint) equivalences are derivable from the judgmental equalities in definitional isomorphisms.

One example of a definitional isomorphism in dependent type theory is the identity equivalence, defined by two copies of the identity function λχ.χ\lambda \chi.\chi and by the fact that the identity function is a definitional involution, where (λχ.χ)((λχ.χ)(x))(\lambda \chi.\chi)((\lambda \chi.\chi)(x)) reduces to xx.

Definitional isomorphisms are used to characterize the identity types of various basic types such as dependent sum types and dependent product types in binary parametric observational type theory and higher observational type theory.

In addition, definitional isomorphisms and definitional isomorphism types allow for easier proofs of the typal congruence rules of various types, since the judgmental equalities allow one to avoid transport hell that comes with the usual proofs of typal congruence rules using weak equivalences. See dependent product type for an example of two sets of such proofs using definitional isomorphisms and weak equivalences respectively, the one using definitional isomorphisms is simpler than the one using weak equivalences.

Definitional isomorphism types

Given types AA and BB, one could define the type of definitional isomorphisms between AA and BB. These are given by the following inference rules:

Formation rule for definitional isomorphism types:

ΓAtypeΓBtypeΓABtype\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \cong B \; \mathrm{type}}

Introduction rule for definitional isomorphism types:

ΓAtypeΓBtypeΓ,x:Af(x):BΓ,y:Bg(y):AΓ,x:Ag(f(x))x:AΓ,y:Bf(g(y))y:BΓtoDefIso(x:A.f(x),y:B.g(y)):AB\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B \vdash g(y):A \quad \Gamma, x:A \vdash g(f(x)) \equiv x:A \quad \Gamma, y:B \vdash f(g(y)) \equiv y:B}{\Gamma \vdash \mathrm{toDefIso}(x:A.f(x), y:B.g(y)):A \cong B}

Elimination rules for definitional isomorphism types:

ΓAtypeΓBtypeΓ,e:AB,x:Ae(x):BΓAtypeΓBtypeΓ,e:AB,y:Be(x):A\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, x:A \vdash \overrightarrow{e}(x):B} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, y:B \vdash \overleftarrow{e}(x):A}
ΓAtypeΓBtypeΓ,e:AB,x:Ae(e(x))x:AΓAtypeΓBtypeΓ,e:AB,y:Be(e(y))y:B\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, x:A \vdash \overleftarrow{e}(\overrightarrow{e}(x)) \equiv x:A} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, y:B \vdash \overrightarrow{e}(\overleftarrow{e}(y)) \equiv y:B}

Computation rules for definitional isomorphism types:

ΓAtypeΓBtypeΓ,x:Af(x):BΓ,y:Bg(y):AΓ,x:Ag(f(x))x:AΓ,y:Bf(g(y))y:BΓ,x:AtoDefIso(x:A.f(x),y:B.g(y))(x)f(x):B\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B \vdash g(y):A \quad \Gamma, x:A \vdash g(f(x)) \equiv x:A \quad \Gamma, y:B \vdash f(g(y)) \equiv y:B}{\Gamma, x:A \vdash \overrightarrow{\mathrm{toDefIso}(x:A.f(x), y:B.g(y))}(x) \equiv f(x):B}
ΓAtypeΓBtypeΓ,x:Af(x):BΓ,y:Bg(y):AΓ,x:Ag(f(x))x:AΓ,y:Bf(g(y))y:BΓ,y:BtoDefIso(x:A.f(x),y:B.g(y))(y)g(y):A\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B \vdash g(y):A \quad \Gamma, x:A \vdash g(f(x)) \equiv x:A \quad \Gamma, y:B \vdash f(g(y)) \equiv y:B}{\Gamma, y:B \vdash \overleftarrow{\mathrm{toDefIso}(x:A.f(x), y:B.g(y))}(y) \equiv g(y):A}

Uniqueness rules for definitional isomorphism types:

ΓAtypeΓBtypeΓ,e:ABtoDefIso(x:A.e(x),y:B.e(y))e:AB\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B \vdash \mathrm{toDefIso}(x:A.\overrightarrow{e}(x), y:B.\overleftarrow{e}(y)) \equiv e:A \cong B}

Properties

In the presence of definitional isomorphism types and the inductively defined identity types, transport can be defined as definitional isomorphism via the J-rule, since the identity function or identity equivalence is a definitional isomorphism and thus one can apply the J-rule to reflexivity to get the identity as a definitional isomorphism:

x:A,y:A,p:x= Ayind Id A,B(x)B(y)(p):B(x)B(y)x:A, y:A, p:x =_A y \vdash \mathrm{ind}_{\mathrm{Id}}^{A, B(x) \cong B(y)}(p):B(x) \cong B(y)
x:Aind Id A,B(x)B(x)(refl A(x))toDefIso(χ:A.χ,χ:A.χ):B(x)B(x)x:A \vdash \mathrm{ind}_{\mathrm{Id}}^{A, B(x) \cong B(x)}(\mathrm{refl}_A(x)) \equiv \mathrm{toDefIso}(\chi:A.\chi, \chi:A.\chi):B(x) \cong B(x)

However, the dependent type theory will no longer have decidable equality if it has definitional isomorphism types, as definitional isomorphism types allow one to include arbitrary definitional isomorphisms in contexts. In turn, this allows one to define fixed point operators, which are incompatible with decidable equality for dependent type theories.

References

Definitional isomorphism is called “definitional section-retraction” in:

The proof assistant Narya makes use of definitional isomorphisms to characterize identity/bridge types.

Last revised on June 16, 2024 at 18:49:13. See the history of this page for a list of all contributions to it.