From: Michael Shulman <firstname.lastname@example.org> To: Valery Isaev <email@example.com> Cc: Jon Sterling <firstname.lastname@example.org>, "HomotopyTypeTheory@googlegroups.com" <email@example.com> Subject: Re: [HoTT] New theorem prover Arend is released Date: Sun, 11 Aug 2019 08:39:46 -0400 Message-ID: <CAOvivQxMwmkA0o2XBecGQaiaVFO0rjwKvZfbMqS+e363GJwRbA@mail.gmail.com> (raw) In-Reply-To: <CAA520ftYLYwiZs0B3fmuYb==8mWiOpVD0yVbV8otfTEfgWV8UQ@mail.gmail.com> On Sun, Aug 11, 2019 at 6:47 AM Valery Isaev <firstname.lastname@example.org> wrote: > I don't remember well, but I think the idea is that you need to prove that there is a trivial cofibration Eq(A,B) -> F(U^I,A,B), where the first object is the object of equivalences between A and B and the second object is the fiber of U^I over A and B. The fact that this map is a weak equivalence is just the univalence axiom. The problem is to show that it is a cofibration and whether this is true or not depends on the definition of Eq(A,B). I don't actually remember whether I finished this proof. Yes, in a straightforward approach that's what you would need to prove, or more precisely that the map Eq -> U^I is a trivial cofibration in the slice over UxU (since the lifting has to be done in the universal case rather than just over each global element). Possibly you could get away with less, since you're only asking to recover the action of the given equivalence as a function (rather than the entire equivalence data). Perhaps a clever choice of fibration structure in the equivalence extension property would suffice. > Since F(A,p) is the usual (inductive) data type, you can do everything you can do with other data types. In particular, since it has only one constructor with one parameter A, it is easy to proof that it is equivalent to A. Okay, it sounds like you're saying that the datatype defined with \use\level is F(A,p) and the analogous one defined without \use\level is A, while both have their usual rules but are not identical. In this case you have somewhat *more* than an equivalence between F(A,p) and A, at least if all datatypes have their usual definitional computation rules, since an arbitrary type "F(A,p)" that is only *equivalent* to a datatype will only inherit a typal computation rule. It seems plausible though that one could construct universes of n-types that are strictly closed under a given datatype construction that preserves n-types (at least, assuming one can do the same for arbitrary universes). However, the documentation also suggests that \use\level can be applied to plain definitions (\func). How is F(A,p) distinguished from A in this case? -- 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/CAOvivQxMwmkA0o2XBecGQaiaVFO0rjwKvZfbMqS%2Be363GJwRbA%40mail.gmail.com.
next prev parent reply index Thread overview: 20+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-08-06 22:16 Валерий Исаев 2019-08-07 15:01 ` Andrej Bauer 2019-08-07 22:13 ` Nicolai Kraus 2019-08-08 9:55 ` Valery Isaev 2019-08-10 9:47 ` Michael Shulman 2019-08-10 12:30 ` Valery Isaev 2019-08-10 12:37 ` Valery Isaev 2019-08-08 12:20 ` Jon Sterling 2019-08-08 12:29 ` Bas Spitters 2019-08-08 14:44 ` Valery Isaev 2019-08-08 15:11 ` Jon Sterling 2019-08-08 15:22 ` Valery Isaev 2019-08-10 9:42 ` Michael Shulman 2019-08-10 12:24 ` Valery Isaev 2019-08-10 23:37 ` Michael Shulman 2019-08-11 10:46 ` Valery Isaev 2019-08-11 12:39 ` Michael Shulman [this message] 2019-08-11 16:55 ` Michael Shulman 2019-08-12 14:44 ` Daniel R. Grayson 2019-08-12 17:32 ` Michael Shulman
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