Discussion of Homotopy Type Theory and Univalent Foundations
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From: Valery Isaev <valery.isaev@gmail.com>
To: Jon Sterling <jon@jonmsterling.com>
Cc: "'Martin Escardo' via Homotopy Type Theory"
	<HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] New theorem prover Arend is released
Date: Thu, 8 Aug 2019 18:22:52 +0300
Message-ID: <CAA520fuU-BEcdg6mrAkTxmqpDhG9pHpD-1a=X7ZU-n8q57dKsg@mail.gmail.com> (raw)
In-Reply-To: <d0fd1d18-136b-41fa-b721-f64b9c487376@www.fastmail.com>

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чт, 8 авг. 2019 г. в 18:11, Jon Sterling <jon@jonmsterling.com>:

> Hi Valery (and Bas),
>
> Thanks very much for the reference! Can you say a bit more about the
> interpretation in model categories? I have the following specific questions:
>
> - By "Cartesian model category" do you refer to the pushout-product axiom?
> I apologize for my ignorance, model categories are not my area of expertise.
>
>
Yes.


> - In most models of HoTT that I'm familiar with, including simplicial sets
> and several versions of cubical sets, the interval is not fibrant. What
> interpretation do have in mind for I? (btw, sorry if this has been
> discussed before!).
>

You can take any contractible fibrant object such that the map 1+1 -> I is
a cofibration. Such an object always exists (just factor the map 1+1 -> 1
as a cofibration followed by trivial fibration), so the only additional
assumption is that the exponential (-)^I exists and the pushout-product
axiom holds.


>
> Thanks,
> Jon
>
>
> On Thu, Aug 8, 2019, at 10:45 AM, Valery Isaev wrote:
> > Yes, Arend implements the theory described in this document.
> > Semantically, the additional constructions of this theory correspond to
> > the assumption that the model has a fibrant object I such that maps
> > <id,left> : X -> X \times I have the left lifting property with respect
> > to fibrations, and the path object functor is given by (-)^I. So, the
> > usual interpretation in model categories (and other similar models) of
> > HoTT extends to an interpretation of this theory if the model category
> > is a Cartesian model category.
> >
> > Regards,
> > Valery Isaev
> >
> >
> > чт, 8 авг. 2019 г. в 15:29, Bas Spitters <b.a.w.spitters@gmail.com>:
> > > I imagine it could be related to earlier discussions, but Valery will
> > >  correct me:
> > > https://groups.google.com/forum/#!topic/homotopytypetheory/N8jw_5h2Qjs
> > > https://valis.github.io/doc.pdf
> > >
> > >  On Thu, Aug 8, 2019 at 2:20 PM Jon Sterling <jon@jonmsterling.com>
> wrote:
> > >  >
> > >  > Dear Valery,
> > >  >
> > >  > Arend looks really impressive, especially the IDE features! I look
> forward to trying it. I like the little screen demos on the website.
> > >  >
> > >  > We have been curious for some time if someone could begin to
> explain what type theory Arend implements --- I am not necessarily looking
> for something super precise, but it would be great to have a high-level
> gloss that would help experts in the semantics of HoTT understand where
> Arend's type theory lies. For instance, I can see that Arend uses an
> interval, but this interval seems to work a bit differently from the
> interval in some other type theories. Is there any note or document that
> explains some of the mathematics behind Arend?
> > >  >
> > >  > Nice work! And I look forward to hearing and reading more.
> > >  >
> > >  > Best,
> > >  > Jon
> > >  >
> > >  >
> > >  > On Tue, Aug 6, 2019, at 6:16 PM, Валерий Исаев wrote:
> > >  > > Arend is a new theorem prover that have been developed at
> JetBrains
> > >  > > <https://www.jetbrains.com/> for quite some time. We are proud to
> > >  > > announce that the first version of the language was released! To
> learn
> > >  > > more about Arend, visit our site <https://arend-lang.github.io/>.
> > >  > >
> > >  > > Arend is based on a version of homotopy type theory that includes
> some
> > >  > > of the cubical features. In particular, it has native higher
> inductive
> > >  > > types, including higher inductive-inductive types. It also has
> other
> > >  > > features which are necessary for a theorem prover such as universe
> > >  > > polymorphism and class system. We believe that a theorem prover
> should
> > >  > > be convenient to use. That is why we also developed a plugin for
> > >  > > IntelliJ IDEA <https://www.jetbrains.com/idea/> that turns it
> into a
> > >  > > full-fledged IDE for the Arend language. It implements many
> standard
> > >  > > features such as syntax highlighting, completion, auto import,
> and auto
> > >  > > formatting. It also has some language-specific features such as
> > >  > > incremental typechecking and various refactoring tools.
> > >  > >
> > >  > > To learn more about Arend, you can check out the documentation
> > >  > > <https://arend-lang.github.io/documentation>. You can also learn
> a lot
> > >  > > from studying the standard library
> > >  > > <https://github.com/JetBrains/arend-lib>. It implements some
> basic
> > >  > > algebra, including localization of rings, and homotopy theory,
> > >  > > including joins, modalities, and localization of types.
> > >  > >
> > >  > > Frequently asked questions (that nobody asked):
> > >  > > * Why do we need another theorem prover? We believe that a theorem
> > >  > > prover should be convenient to use. This means that it should
> have an
> > >  > > IDE comparable to that of mainstream programming languages. That
> is why
> > >  > > we implemented IntelliJ Arend
> > >  > > <https://arend-lang.github.io/about/intellij-features>. This
> also means
> > >  > > that the underlying theory should be powerful and expressive.
> That is
> > >  > > why Arend is based on homotopy type theory and has features such
> as an
> > >  > > impredicative type of propositions and a powerful class system.
> > >  > > * Does Arend have tactics? Not yet, but we are working on it.
> > >  > > * Does Arend have the canonicity property, i.e. does it evaluate
> > >  > > closed expressions to their canonical forms? No, but it computes
> more
> > >  > > terms than ordinary homotopy type theory, which makes it more
> > >  > > convenient in many aspects.
> > >  > > If you want to know about language updates, you can follow us on
> > >  > > twitter <https://twitter.com/ArendLang>. Questions, suggestions,
> and
> > >  > > comments are welcome at google groups
> > >  > > <https://groups.google.com/forum/#!forum/arend-lang>.
> > >  > >
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> <
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  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 [this message]
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
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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