nLab type theoretic axiom of choice

 Idea

In dependent type theory, the type theoretic axiom of choice is a version of axiom of choice which does not use any bracket types in the theorem, as it uses the propositions as types interpretation of dependent type theory. Despite its name, the type theoretic axiom of choice is provable in dependent type theory from the inference rules of dependent product types, dependent sum types, and identity types, and so really deserves to be called the type theoretic principle of choice. This is in contrast to what is referred to as the propositional axiom of choice or simply axiom of choice which does use bracket types via the propositions as subsingletons interpretation of dependent type theory and is just the usual axiom of choice expressed in dependent type theory.

Statement

The type theoretic axiom of choice is the statement that given types $A$ and $B$ and type family $x:A, y:B \vdash R(x, y)$, one can construct

$$\mathrm{ac}_R \coloneqq \lambda g.(\lambda x.\pi_1(g(x)), \lambda x.\pi_2(g(x))):\left(\prod_{x:A} \sum_{y:B} R(x, y)\right) \to \left(\sum_{f:A \to B} \prod_{x:A} R(x, f(x))\right)$$

The propositions as types interprets disjunction and existential quantification directly as the sum type and dependent sum type respectively, and the statement of the axiom of choice comes out as simply the statement that products distribute over coproducts. (See distributivity pullback for a discussion in terms of the internal type theory of a locally cartesian closed category.)

The equivalent form of the axiom of choice involving cartesian products of inhabited types then becomes in type theory the statement that “any dependent product of any family of pointed types is pointed”. and is just the identity function on the dependent product type.

Related concepts

• axiom of choice

• propositional axiom of choice

 References

The type theoretic axiom of choice is found in:

• Univalent Foundations Project, section 1.6, of Homotopy Type Theory – Univalent Foundations of Mathematics