[HoTT] Definitional equality of types as strict bijection

[HoTT] Definitional equality of types as strict bijection

[Madeleine Birchfield]

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In this thread, I follow the approach in Egbert Rijke's textbook in having 
a separate type judgment, rather than the HoTT book's judgment in having an 
infinite tower of universes and no separate type judgment, and I follow 
Agda in using the bare equals sign for definitional equality. 

In structural set theories like William Lawvere's ETCS or Mike Shulman's 
SEAR it is quite common to have equality only for the sorts of elements, 
functions, and relations, and not have any primitive equality relation for 
the sort of sets. This means that the only way to compare sets for 
"equality" is through a bijection of sets. 

In addition, Mike Shulman's talks on Higher Observational Type Theory back 
in May of 2022 includes a section where he talks about the right notion of 
"definitional univalence" is not using the usual notion of definitional 
equality of types, but rather strict bijection, where there are functions 
idtoequiv(a, b):Id_U(a, b) -> Equiv_U(a, b)
equivtoid(a, b):Equiv_U(a, b) -> Id_U(a, b) 
between the identity types Id_U(a, b) and the type of equivalences 
Equiv_U(a, b) of elements a:U and b:U of a universe U, and definitional 
equalities of elements
equivtoid(a, b)(idtoequiv(a, b)(p)) = p:Id_U(a, b)
idtoequiv(a, b)(equivtoid(a, b)(R)) = R:Equiv_U(a, b)

Using insights from structural set theory, one could go beyond definitional 
univalence in Mike Shulman's Higher Observational Type Theory and directly 
modify the structural rules for definitional equality of types so that it 
behaves not as an equivalence relation a la first order logic with 
equality, but rather strict bijection of types a la structural set theory, 
using the following rules:

Gamma |- A = B type
---------------------------------
Gamma, x:A |- f(x):B

Gamma |- A = B type
---------------------------------
Gamma, y:B |- g(y):A

Gamma |- A = B type
---------------------------------
Gamma, x:A |- g(f(x)) = x:A

Gamma |- A = B type
---------------------------------
Gamma, y:B |- f(g(y)) = y:B

The variable conversion rule in dependent type theory then becomes 
substitution of g(y) for x, 

Gamma |- A = A' type    Gamma, x:A, Delta(x) |- J(x)
------------------------------------------------------------------------------------
Gamma, y:A', Delta(g(y)) |- J(g(y))

The first congruence rule for substitution is the following rule

Gamma |- a = a':A    Gamma, x:A, Delta(x) |- B(x) type
------------------------------------------------------------------------------------
Gamma, Delta(a) |- B(a) = B(a') type

and the strict bijection rules for definitional equality of types implies 
that f and g for the definitional equality B(a) = B(a') type are judgmental 
versions of transport. The second congruence rule for substitution becomes 
merely a judgmental version of application to paths:

Gamma |- a = a':A    Gamma, x:A, Delta(x) |- b(x):B(x) type
------------------------------------------------------------------------------------------
Gamma, Delta(a) |- b(a) = g(b(a')):B(a)

Definitional univalence then becomes under this collection of structural 
rules the following rule

Gamma |- U type    Gamma |- A:U   Gamma |- B:U
------------------------------------------------------------------------------------------
Gamma |- Id_U(A, B) = Eq_U(A, B) type

from which one could derive Mike Shulman's original 4 rules for 
definitionally univalent universes. 

Madeleine Birchfield

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