[HoTT] Identity types of types, and univalence for the entire type theory

[Madeleine Birchfield <madeleinebirchfield@gmail.com>]

[-- Attachment #1.1: Type: text/plain, Size: 5276 bytes --]

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

-- 
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/3ff0b19b-0db2-4b63-b75d-3da9101a2f96n%40googlegroups.com.

[-- Attachment #1.2: Type: text/html, Size: 5605 bytes --]