nLab definitional isomorphism

Definition

In dependent type theory, a definitional isomorphism or judgmental isomorphism between two types $A$ and $B$ consists of functions $f:A \to B$ and $f^{-1}:B \to A$ such that given any term $x:A$, the term $f^{-1}(f(x))$ is judgmentally equal to $x$ and given any term $y:B$, the term $f(f^{-1}(y))$ is judgmentally equal to $y$. In symbols,

$$x:A \vdash f^{-1}(f(x)) \equiv x:A \quad \mathrm{and} \quad y:B \vdash f(f^{-1}(y)) \equiv y:B$$

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 $\lambda \chi.\chi$ and by the fact that the identity function is a definitional involution, where $(\lambda \chi.\chi)((\lambda \chi.\chi)(x))$ reduces to $x$.

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 $A$ and $B$, one could define the type of definitional isomorphisms between $A$ and $B$. These are given by the following inference rules:

Formation rule for definitional isomorphism types:

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

Introduction rule for definitional isomorphism types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B \vdash g(y):A \quad \Gamma, x:A \vdash g(f(x)) \equiv x:A \quad \Gamma, y:B \vdash f(g(y)) \equiv y:B}{\Gamma \vdash \mathrm{toDefIso}(x:A.f(x), y:B.g(y)):A \cong B}$$

Elimination rules for definitional isomorphism types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, x:A \vdash \overrightarrow{e}(x):B} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, y:B \vdash \overleftarrow{e}(x):A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, x:A \vdash \overleftarrow{e}(\overrightarrow{e}(x)) \equiv x:A} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, e:A \cong B, y:B \vdash \overrightarrow{e}(\overleftarrow{e}(y)) \equiv y:B}$$

Computation rules for definitional isomorphism types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B \vdash g(y):A \quad \Gamma, x:A \vdash g(f(x)) \equiv x:A \quad \Gamma, y:B \vdash f(g(y)) \equiv y:B}{\Gamma, x:A \vdash \overrightarrow{\mathrm{toDefIso}(x:A.f(x), y:B.g(y))}(x) \equiv f(x):B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B \quad \Gamma, y:B \vdash g(y):A \quad \Gamma, x:A \vdash g(f(x)) \equiv x:A \quad \Gamma, y:B \vdash f(g(y)) \equiv y:B}{\Gamma, y:B \vdash \overleftarrow{\mathrm{toDefIso}(x:A.f(x), y:B.g(y))}(y) \equiv g(y):A}$$

Uniqueness rules for definitional isomorphism types:

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

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:

$$x:A, y:A, p:x =_A y \vdash \mathrm{ind}_{\mathrm{Id}}^{A, B(x) \cong B(y)}(p):B(x) \cong B(y)$$

$$x:A \vdash \mathrm{ind}_{\mathrm{Id}}^{A, B(x) \cong B(x)}(\mathrm{refl}_A(x)) \equiv \mathrm{toDefIso}(\chi:A.\chi, \chi:A.\chi):B(x) \cong B(x)$$

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.

Related concepts

• judgmental equality
• function, function type
• equivalence of types, equivalence type
• isomorphism

References

• Emily Riehl, Mike Shulman, A type theory for synthetic ∞-categories, Higher Structures 1 1 (2017) [arxiv:1705.07442, published article]

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