From: "Rafaël Bocquet" <email@example.com>
Subject: [HoTT] Frobenius eliminators
Date: Thu, 25 Jul 2019 11:58:26 +0200 [thread overview]
Message-ID: <firstname.lastname@example.org> (raw)
[-- Attachment #1: Type: text/plain, Size: 3459 bytes --]
This is related to a previous discussion on this mailing list :
"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
I consider roughly the same framework as in "The homotopy theory of type
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
- 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 ?
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 HomotopyTypeTheoryemail@example.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/a20122bc-289b-4fce-1183-76b66aed395f%40ens.fr.
[-- Attachment #2: Type: text/html, Size: 4718 bytes --]
reply other threads:[~2019-07-25 9:58 UTC|newest]
Thread overview: [no followups] expand[flat|nested] mbox.gz Atom feed
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).