From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: X-Spam-Checker-Version: SpamAssassin 3.4.2 (2018-09-13) on inbox.vuxu.org X-Spam-Level: X-Spam-Status: No, score=-1.2 required=5.0 tests=DKIM_SIGNED,DKIM_VALID, DKIM_VALID_AU,DKIM_VALID_EF,FREEMAIL_FORGED_FROMDOMAIN,FREEMAIL_FROM, HEADER_FROM_DIFFERENT_DOMAINS,HTML_MESSAGE,MAILING_LIST_MULTI, RCVD_IN_DNSWL_NONE autolearn=ham autolearn_force=no version=3.4.2 Received: from mail-ot1-x33a.google.com (mail-ot1-x33a.google.com [IPv6:2607:f8b0:4864:20::33a]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id d7fb50b6 for ; Sat, 25 May 2019 13:34:58 +0000 (UTC) Received: by mail-ot1-x33a.google.com with SMTP id 68sf6034286otu.18 for ; Sat, 25 May 2019 06:34:57 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:date:from:to:message-id:in-reply-to:references:subject :mime-version:x-original-sender:precedence:mailing-list:list-id :list-post:list-help:list-archive:list-unsubscribe; bh=cWFIvuSntGbasjgi/enr+BclsBMMkS2o55s45mK45Vo=; b=sLl6ogkDAH5zR8oDuYmtWaWiZD36oxm1iy3q3tUl2cKNJe4gGrmHPUgun5SDLdh0Ke AEppED9taxxRoZqFUOtVdHtX3TTkh35BuF0y4iWqPpxPhrfsthVQ5okfSfrRhawmRQ83 QqISTWg6NBqHFNc87Ul08Phkzo1ONOdgoiOyDU656BV3+JQ5mIrgHWay5Beu/1V/9MwZ okpeDDB1SZDCKyduF7hxi7gTVLZHJXUu9xvRD6g70xZzhXn+OTA22Mkl1KKTIJMd6pVE PXhLhiRcfhP90xHLxLfJriY14xrRiZ02eukFlrQV9Feo2qrhNIxGZi2hHJsfDDOeDa5w WEqw== DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=date:from:to:message-id:in-reply-to:references:subject:mime-version :x-original-sender:precedence:mailing-list:list-id:list-post :list-help:list-archive:list-unsubscribe; bh=cWFIvuSntGbasjgi/enr+BclsBMMkS2o55s45mK45Vo=; b=NG6cPhY1XRkgIoQne0erMuQ6F4PMTA+xSzDmyK5bX72rTT87/ZxvEDi0XhqYExC/Mg a+Ldf4em4MAKP6gTK1jhoOmvGuTPsXXQ3oJe0WvVJozrNe2U5tUgwA7VsGQtqh0KHKxr 6G+UjoKwuiQMLE0k3ejUpjBfiw0DVuEK3EiP1yOUhoWZCsXNqC/7loeUPEO3u9WU54rG 2RM2WcMHvIgPVsW7H1l5HHTpREtALlLFic/4dWUX4IP2eqYSzQOS05hRgS9H5CZtZNbr VhXKeK7BN1uaAjqI8t8qfwELo1dhJZF7mxR03KXtppGnd2Q0IJL44mn49PTFvAJplLtL TpWA== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=sender:x-gm-message-state:date:from:to:message-id:in-reply-to :references:subject:mime-version:x-original-sender:precedence :mailing-list:list-id:x-spam-checked-in-group:list-post:list-help :list-archive:list-unsubscribe; bh=cWFIvuSntGbasjgi/enr+BclsBMMkS2o55s45mK45Vo=; b=GWPp4zaY6Jji+uWGBkQB8aMR1Sz5BPyO/s5KvI9STON62RFzqWamBvZ7IT1BzH4SCU m45FGvG6aiXHjx1Dj6sQI1Ihio9csZTrrZe1Sxg7fgyFncUCSFv6f74pFOHHTBxG3IaA SmLZLnSaugc089pqYDXTD91fpKzCt0rhuT1s3ITlgwau7SYJj7QZucEnu68Ic4AkLFCm DBUTlpNGAIZ1Pw6H2LDhd4t5X1eYxBT+R8CSlPSJdKp65Hzp69xEavYD1s3vsho6nCLN mZW90Ud3VaEM8f4/0i2lZBE+p1KZ2GN4XrYcGVCrIDr21p92Tdcas6URwm2QMeNM/anV W8mg== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAWFzseNFdx0WmaXKd7vgMebNnk/Bz5xtGtjsqR6N6VsHcj83taE dMXLQp3aR6bW8O4kFkZ2bqE= X-Google-Smtp-Source: APXvYqz8dRZcguDgv+p71pApOTEvjn63FlUM2SVQ83zirI6p46F7B1XJEO8s56vhrNInoo55ngUzMA== X-Received: by 2002:a9d:630f:: with SMTP id q15mr143224otk.21.1558791296712; Sat, 25 May 2019 06:34:56 -0700 (PDT) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:a05:6830:14f:: with SMTP id j15ls2236856otp.9.gmail; Sat, 25 May 2019 06:34:56 -0700 (PDT) X-Received: by 2002:a05:6830:1597:: with SMTP id i23mr4853otr.281.1558791296240; Sat, 25 May 2019 06:34:56 -0700 (PDT) Date: Sat, 25 May 2019 06:34:55 -0700 (PDT) From: Juan Ospina To: Homotopy Type Theory Message-Id: <18681ec4-dc8d-42eb-b42b-b9a9f639d89e@googlegroups.com> In-Reply-To: References: Subject: Re: [HoTT] doing "all of pure mathematics" in type theory MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_1358_1755135488.1558791295485" X-Original-Sender: jospina65@gmail.com Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , ------=_Part_1358_1755135488.1558791295485 Content-Type: multipart/alternative; boundary="----=_Part_1359_572354898.1558791295485" ------=_Part_1359_572354898.1558791295485 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable On page 117 of https://arxiv.org/pdf/1808.10690.pdf appears the "additivity= =20 axiom". Please let me know if the following formulation of the such axiom= =20 is correct: [image: additivity.jpg] On Saturday, May 25, 2019 at 5:22:41 AM UTC-5, awodey wrote: > > A useful example for you might be Floris van Doorn=E2=80=99s formalizatio= n of=20 > the Atiyah-Hirzebruch and Serre spectral sequences for cohomology=20 > in HoTT using Lean:=20 > > https://arxiv.org/abs/1808.10690=20 > > Regards,=20 > > Steve=20 > > > On May 25, 2019, at 12:12 PM, Kevin Buzzard > wrote:=20 > >=20 > > Hi from a Lean user.=20 > >=20 > > As many people here will know, Tom Hales' formal abstracts project=20 > https://formalabstracts.github.io/ wants to formalise many of the=20 > statements of modern pure mathematics in Lean. One could ask more general= ly=20 > about a project of formalising many of the statements of modern pure=20 > mathematics in an arbitrary system, such as HoTT. I know enough about the= =20 > formalisation process to know that whatever system one chooses, there wil= l=20 > be pain points, because some mathematical ideas fit more readily into som= e=20 > foundational systems than others.=20 > >=20 > > I have seen enough of Lean to become convinced that the pain points=20 > would be surmountable in Lean. I have seen enough of Isabelle/HOL to beco= me=20 > skeptical about the idea that it would be suitable for all of modern pure= =20 > mathematics, although it is clearly suitable for some of it; however it= =20 > seems that simple type theory struggles to handle things like tensor=20 > products of sheaves of modules on a scheme, because sheaves are dependent= =20 > types and it seems that one cannot use Isabelle's typeclass system to=20 > handle the rings showing up in a sheaf of rings.=20 > >=20 > > I have very little experience with HoTT. I have heard that the fact tha= t=20 > "all constructions must be isomorphism-invariant" is both a blessing and = a=20 > curse. However I would like to know more details. I am speaking at the Bi= g=20 > Proof conference in Edinburgh this coming Wednesday on the pain points=20 > involved with formalising mathematical objects in dependent type theory a= nd=20 > during the preparation of my talk I began to wonder what the analogous=20 > picture was with HoTT.=20 > >=20 > > Everyone will have a different interpretation of "modern pure=20 > mathematics" so to fix our ideas, let me say that for the purposes of thi= s=20 > discussion, "modern pure mathematics" means the statements of the theorem= s=20 > publishsed by the Annals of Mathematics over the last few years, so for= =20 > example I am talking about formalising statements of theorems involving= =20 > L-functions of abelian varieties over number fields, Hodge theory,=20 > cohomology of algebraic varieties, Hecke algebras of symmetric groups,=20 > Ricci flow and the like; one can see titles and more at=20 > http://annals.math.princeton.edu/2019/189-3 . Classical logic and the=20 > axiom of choice are absolutely essential -- I am only interested in the= =20 > hard-core "classical mathematician" stance of the way mathematics works,= =20 > and what it is.=20 > >=20 > > If this is not the right forum for this question, I would be happily=20 > directed to somewhere more suitable. After spending 10 minutes failing to= =20 > get onto ##hott on freenode ("you need to be identified with services") I= =20 > decided it was easier just to ask here. If people want to chat directly I= =20 > am usually around at https://leanprover.zulipchat.com/ (registration=20 > required, full names are usually used, I'll start a HoTT thread in=20 > #mathematics).=20 > >=20 > > Kevin Buzzard=20 > >=20 > > --=20 > > You received this message because you are subscribed to the Google=20 > Groups "Homotopy Type Theory" group.=20 > > To unsubscribe from this group and stop receiving emails from it, send= =20 > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com = .=20 > > > To view this discussion on the web visit=20 > https://groups.google.com/d/msgid/HomotopyTypeTheory/a57315f6-cbd6-41a5-a= 3b7-b585e33375d4%40googlegroups.com.=20 > > > For more options, visit https://groups.google.com/d/optout.=20 > > --=20 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 e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/= HomotopyTypeTheory/18681ec4-dc8d-42eb-b42b-b9a9f639d89e%40googlegroups.com. For more options, visit https://groups.google.com/d/optout. ------=_Part_1359_572354898.1558791295485 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
On page 117 of https://arxiv.org/pdf/1808.10690.pdf a= ppears the "additivity axiom".=C2=A0 Please let me know if the fo= llowing formulation of the such axiom is correct:

3D"additivity.jpg"



On Saturday, May 25, 2019 at 5:22:41 = AM UTC-5, awodey wrote:
A usefu= l example for you might be Floris van Doorn=E2=80=99s formalization of=20
the Atiyah-Hirzebruch and Serre spectral sequences for cohomology=20
in HoTT using Lean:

=C2=A0https://arxiv.org/abs/1808.10690

Regards,

Steve

> On May 25, 2019, at 12:12 PM, Kevin Buzzard <kevin....@gmail.com= > wrote:
>=20
> Hi from a Lean user.=20
>=20
> As many people here will know, Tom Hales' formal abstracts pro= ject https://formalabstracts.github.io/ wants = to formalise many of the statements of modern pure mathematics in Lean. One= could ask more generally about a project of formalising many of the statem= ents of modern pure mathematics in an arbitrary system, such as HoTT. I kno= w enough about the formalisation process to know that whatever system one c= hooses, there will be pain points, because some mathematical ideas fit more= readily into some foundational systems than others.
>=20
> I have seen enough of Lean to become convinced that the pain point= s would be surmountable in Lean. I have seen enough of Isabelle/HOL to beco= me skeptical about the idea that it would be suitable for all of modern pur= e mathematics, although it is clearly suitable for some of it; however it s= eems that simple type theory struggles to handle things like tensor product= s of sheaves of modules on a scheme, because sheaves are dependent types an= d it seems that one cannot use Isabelle's typeclass system to handle th= e rings showing up in a sheaf of rings.
>=20
> I have very little experience with HoTT. I have heard that the fac= t that "all constructions must be isomorphism-invariant" is both = a blessing and a curse. However I would like to know more details. I am spe= aking at the Big Proof conference in Edinburgh this coming Wednesday on the= pain points involved with formalising mathematical objects in dependent ty= pe theory and during the preparation of my talk I began to wonder what the = analogous picture was with HoTT.
>=20
> Everyone will have a different interpretation of "modern pure= mathematics" so to fix our ideas, let me say that for the purposes of= this discussion, "modern pure mathematics" means the statements = of the theorems publishsed by the Annals of Mathematics over the last few y= ears, so for example I am talking about formalising statements of theorems = involving L-functions of abelian varieties over number fields, Hodge theory= , cohomology of algebraic varieties, Hecke algebras of symmetric groups, Ri= cci flow and the like; one can see titles and more at http://annals.math.princeton.edu/2019/189-3<= /a> . Classical logic and the axiom of choice are absolutely essential -- I= am only interested in the hard-core "classical mathematician" st= ance of the way mathematics works, and what it is.
>=20
> If this is not the right forum for this question, I would be happi= ly directed to somewhere more suitable. After spending 10 minutes failing t= o get onto ##hott on freenode ("you need to be identified with service= s") I decided it was easier just to ask here. If people want to chat d= irectly I am usually around at https://leanprover.zulipchat.= com/ (registration required, full names are usually used, I'll= start a HoTT thread in #mathematics).
>=20
> Kevin Buzzard
>=20
> --=20
> 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 h= ttps://groups.google.com/d/msgid/HomotopyTypeTheory/a57315f6-cbd6= -41a5-a3b7-b585e33375d4%40googlegroups.com.
> For more options, visit https://groups.go= ogle.com/d/optout.

--
You received this message because you are subscribed to the Google Groups &= quot;Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an e= mail to = HomotopyTypeTheory+unsubscribe@googlegroups.com.
To view this discussion on the web visit https://groups.google.co= m/d/msgid/HomotopyTypeTheory/18681ec4-dc8d-42eb-b42b-b9a9f639d89e%40googleg= roups.com.
For more options, visit http= s://groups.google.com/d/optout.
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