Discussion of Homotopy Type Theory and Univalent Foundations
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* [HoTT] Frobenius eliminators
@ 2019-07-25  9:58 Rafaël Bocquet
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From: Rafaël Bocquet @ 2019-07-25  9:58 UTC (permalink / raw)
  To: HomotopyTypeTheory

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This is related to a previous discussion on this mailing list : 
https://groups.google.com/d/msg/homotopytypetheory/b96TBt0Bn7Y/IiAcnUL2AQAJ, 
"What is known and/or written about “Frobenius eliminators”?".

I noticed recently that the Frobenius eliminator for identity types is 
related to the notion of strong logical equivalence (the trivial 
fibrations of the left semi-model structure on the category of CwAs with 
Sigma and Id introduced in https://arxiv.org/abs/1610.00037,"The 
homotopy theory of type theories").

The Frobenius, Paulin-Mohring variant of the J eliminator is presented 
by the following inference rule:
   G |- A type   G |- x : A
   G, y:A, p:Id(x,y) |- B(y,p) type*
   G, y:A, p:Id(x,y), b:B(y,p) |- C(y,p,b) type
   G, b:B(x,refl(x)) |- c : C(x,refl(x),b) type
   G |- y : A    G |- p : Id(x,y)    G |- b:B(y,p)
--------------------------------------------------------
   G |- J(A,x,B,C,c,y,p,b) : C(y,p,b)

(where "B(y,p) type*" means that B(y,p) is a finite sequence of types 
over "G, y:A, p:Id(x,y)", i.e. a context in the contextual slice over 
"G, y:A, p:Id(x,y)")

(In the discussion linked above, Valery Isaev gives a proof that if the 
type theory includes sigma types, the "simple" Paulin-Mohring eliminator 
is strong enough to derive the "Frobenius" variant. This is also proven 
by Paige Randall North in https://arxiv.org/abs/1901.03567, in a more 
categorical setting.)

I consider roughly the same framework as in "The homotopy theory of type 
theories".

Recall that a contextual morphism F : C --> D is a strong logical 
equivalence if it satisfies term lifting and type lifting conditions:
- type lifting: for every context G in C, F_G : Ty_C(G) -> Ty_D(F(G)) is 
surjective.
- term lifting: for every context G in C and type A over G, F_A : 
Ter_C(G,A) -> Ter_D(F(G),F(A)) is surjective.

Let C be a CwA with Id and refl. The J eliminator for some pointed type 
(A,x) in some context G can be seen as the statement that the contextual 
morphism F : C[G,y:A,p:Id(x,y)] --> C[G] (where C[G] is a notation for 
the contextual slice over G), sending (y,p) to (x,refl(x)), satisfies 
the term lifting condition. Indeed, the input of the term lifting 
condition is the data of B, C and c, and the output is a term J in the 
context G,y:A,p:Id(x,y),b:B(y,p) of type C(y,p,b), such that F(J) = c.

The type lifting condition can then be derived, assuming that C does not 
include weird type equalities (for instance, when C is cofibrant/freely 
generated, or when C is equipped with a hierarchy of universes ensuring 
that every type belongs to a unique universe).

If we only require F to be a weak logical equivalence (a weak 
equivalence in the left semi-model structure), then we obtain 
propositional identity types. I think that asking for F to be an 
isomorphism forces the identity types to become extensional.

Has this been studied before ? Can this be extended to the elimination 
rules of other type formers ?

Best regards,
Rafaël Bocquet

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