I've had an idea that's been bouncing around in my head for a few days, and I'd thought I'd share it with the rest of the community to see if it it makes sense or if there are any flaws in my thought process.
We work in Martin-Löf type theory with a separate judgment 'A type' for types, as well as weakly Tarski universes consisting of a type U and a type family El. The latter means that the type reflection of the internal types is only equivalent to the corresponding type outside the universe: for example, the type reflection of the internal type of natural numbers N_U : U is only equivalent to the type N, El(N_U) ≅ N, rather than definitionally equal to N. In weakly Tarski universes, we define a type A to be U-small if there exists a term A_U : U with an equivalence small(A, A_U):El(A_U) ≅ A.
The univalence axiom on weakly Tarski universes than implies that the identity type A =_U B is U-small for all A:U and B:U. This in turn implies that in any weakly Tarski universe, it is consistent that there is a function ='_U:(U × U) -> U and an equivalence small(A ='_U B):T(A ='_U B) ≅ (A =_U B) for all A:U and B:U. This means that the univalence axiom could be internalized inside of the the universe U itself as ua:T((A ='_U B) ≅_U (A ≅_U B)), and type reflection implies that the type T((A ='_U B) ≅_U (A ≅_U B)) ≅ (A =_U B ≅ T(A ≅_U B)) is pointed for all A:U and B:U.
On the level of the type theory itself, one should thus be able to add another type former to the entirety of the type theory: the identity types of types A = B, with the following rules:
- The formation rule says that given a judgment 'A type' in context Gamma and a judgment 'B type' in context Gamma, one could form the conclusion 'A = B type' in context Gamma.
- The introduction rule says that given a judgment 'A type' in context Gamma, one could form the conclusion 'refl_A:A = A' in context Gamma.
- The elimination rule says that given a judgment 'A type' in context Gamma, a judgment 'B type' in context Gamma, a judgment 'C(p) type' in the context 'Gamma, p : A = B', and a judgment 't:C(refl_A)' in the context 'Gamma', one could form the conclusion 'J(t, p): C(p)' in the context 'Gamma'
- The computation rule says that given a judgment 'A type' in context Gamma, a judgment 'B type' in context
Gamma, a judgment 'C(p) type' in the context 'Gamma, p : A = B', and a
judgment 't:C(refl_A)' in the context 'Gamma', one could form the conclusion 'J(t, refl_A) ≡ t'' in the context 'Gamma'.
The type theory has two notions of propositional equality which all have an inductive definition, one for the identity types of terms of a type (a =_A b for a:A and b:A), and a second for the identity types of types (A = B for A type and B type).
Provided all of the above is correct, this allows us to define univalence in the entire type theory without using universes at all. Given types A and B, one could inductively define the function idtoequiv_{A, B}:(A = B) -> (A ≅ B) by idtoequiv_{A, A}(refl_A) := id_A. The univalence axiom then says that one could form the term ua_{A, B}:isEquiv(idtoequiv_{A, B})) for whatever definition of isEquiv is used.
In addition, since there are two identity types, there are now two notions of UIP and axiom K: one for the identity types of terms and one for the identity types of types:
- UIP for terms is the rule that given the judgments 'A type', 'a:A', 'b:A', 'p:a =_A b', 'q:a =_A b' all in context Gamma, one could form the conclusion 'uip(A, a, b, p, q):p =_{a =_A b} q'
- UIP for types is the rule that given the judgments 'A type', 'B type', 'p:A = B', 'q:A = B' all
in context Gamma, one could form the conclusion 'uip(A, B, p, q):p
=_{A = B} q'
- axiom K for terms is the rule that given the judgments 'A type', 'a:A', 'p:a =_A a' all
in context Gamma, one could form the conclusion 'uip(A, a, p):p
=_{a =_A a} refl_A(a)'
- axiom K for types is the rule that given the judgments 'A type', 'p:A = A' all
in context Gamma, one could form the conclusion 'uip(A, p):p
=_{A = A} refl_A'
Assuming univalence for the type theory, UIP or axiom K for the identity types of terms implies that every type is a set as usual, but does not necessarily imply UIP or axiom K for the identity types of types. But UIP or axiom K for the identity types of types implies that every type is a proposition and thus a set, and thus implies UIP and axiom K for the identity types of types. However, both sets of axioms of UIP and axiom K are still inconsistent with the existence of a univalent Tarski universe in the type theory, even without univalence for the entire type theory.
Madeleine Birchfield
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