From: "Martín Hötzel Escardó" <escardo.martin@gmail.com> To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com> Subject: Re: [HoTT] Injective types Date: Tue, 7 May 2019 15:06:20 -0700 (PDT) Message-ID: <6bae18da-1ecc-4633-a565-9df222140d87@googlegroups.com> (raw) In-Reply-To: <49d95599-153c-d641-4ae9-a3f7c02b1cc3@gmail.com> [-- Attachment #1.1: Type: text/plain, Size: 3219 bytes --] On 07/05/2019 14:51, Andreas Nuyts wrote: > Even with cumulativity, that sounds suspicious, because cumulativity of > U in a bigger universe V does not obviously give you a map > > (A B : U) -> Id V A B -> Id U A B. > > The J-rule does not allow you to build this map. Indeed. In the non-cumulative system, with V bigger than U, you have a map f : U → V with f A ≃ A. For example, if C is any contractible type in V, you can take f A := A × C. If both U and V are univalent, then any map f : U → V with f A ≃ A is an embedding for all A : U, meaning that the canonical map Id U A B → Id V (f A) (f B) is equivalence (or equivalently that the fibers of f are propositions). I checked this in Agda. The reformulation of your statement above with an explicit inclusion f : U → V, namely (A B : U) -> Id V (f A) (f B) -> Id U A B, which amounts to the left-cancellability of f, is a consequence of f being an embedding, and in general strictly weaker than f being an embedding. But I don't see how to prove even the left-cancellability of f without assuming univalence in the smaller universe U. Assuming that V is univalent, Id V (f A) (f B) gives f A ≃ f B, and then composing with the equivalences A ≃ f A and f B ≃ B we get A ≃ B. If U were univalent we would get Id U A B and hence f would be left-cancellable. But, as I said, the univalence of U and V gives more, namely that f is an embedding. And if V is univalent and f is an embedding, then U is univalent, for A ≃ B is equivalent to f A ≃ f B and hence to Id V (f A) (f B), and hence to Id U A B, using the embedding property in the last step. So if V is univalent, then the smaller universe U is univalent iff every map f : U → V with f A ≃ A for all A : U is an embedding (iff some given such map is an embedding). Martin > > On 7/05/19 14:42, Nils Anders Danielsson wrote: >> On 02/05/2019 22.46, escardo.martin@gmail.com wrote: >>> I can confirm from a 26k line Agda development (with comments and >>> repeated blank lines removed in this counting of the number of lines) >>> that not once did I need to embed a universe into a larger universe, >>> except when I wanted to state the theorem that any universe is a >>> retract of any larger universe if one assumes the propositional >>> resizing axiom (any proposition in a universe U has an equivalent copy >>> in any universe V). So I would say that such situations are *extremely >>> rare* in practice. >> >> I once wrote some code where I made use of univalence at three different >> levels. Does anyone know if one can prove that univalence at one level >> implies univalence at lower levels, without making use of cumulativity? -- 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/6bae18da-1ecc-4633-a565-9df222140d87%40googlegroups.com. For more options, visit https://groups.google.com/d/optout. [-- Attachment #1.2: Type: text/html, Size: 4371 bytes --]

prev parent reply indexThread overview:11+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-04-30 23:05 escardo.martin 2019-05-01 2:50 ` Michael Shulman 2019-05-01 6:25 ` escardo.martin 2019-05-01 16:55 ` Michael Shulman 2019-05-02 20:46 ` escardo.martin 2019-05-03 11:45 ` Michael Shulman 2019-05-03 13:25 ` Kenji Maillard 2019-05-03 18:23 ` Thierry Coquand 2019-05-07 12:42 ` Nils Anders Danielsson 2019-05-07 13:51 ` Andreas Nuyts2019-05-07 22:06 ` Martín Hötzel Escardó [this message]

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