Discussion of Homotopy Type Theory and Univalent Foundations
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From: Gabriel Scherer <gabriel.scherer@gmail.com>
To: Andrej Bauer <andrej.bauer@andrej.com>
Cc: "HomotopyTypeTheory@googlegroups.com"
	<homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] HoTT combinatorics
Date: Mon, 1 Jul 2019 13:52:55 +0200
Message-ID: <CAPFanBECYaoV-WoC8A67LE749=ZfCQWs+SyN9G7mnSvpU6nOfw@mail.gmail.com> (raw)
In-Reply-To: <CAB0nkh2h_L_9ANAZdiu+ZmAXRTi_S7HQDG9RYCuvYbbVk-HmqA@mail.gmail.com>

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Not HoTT-specific, some of the recent type/proof-theory work of Noam
Zeilberger may pique the interest of combinatorists. For example,

  A sequent calculus for the Tamari order
  Noam Zeilberger, 2019
  https://arxiv.org/abs/1803.10080

uses a sequent calculus to describe the structure of an order relation on
trees that combinatorist have studied, and re-derive counting results on
the intervals of that order.

(Reasoning on the structure of an order relation is not too far from the
"higher" concerns of HoTT and higher category-theory.
And the idea of using adequate proof structures to prove a coherence result
has been used in many other more categorical settings.)

On Mon, Jul 1, 2019 at 1:36 PM Andrej Bauer <andrej.bauer@andrej.com> wrote:

> In a couple of days I am gonig to give a talk about HoTT in front of
> 250 combinatorialists at http://fpsac2019.fmf.uni-lj.si
>
> I have some ideas about how to explain that HoTT is relevant to a
> mathematician who studies "simple finite objects", but I'd be
> interested to hear if anyone has anything else to say. I'll gladly
> acknowledge good ideas.
>
> My current plan is to discuss, after a suitable introduction:
>
> 1. The difference between Σ and ∃ is the difference between "explicit
> construction" and "abstract proof of existence".
>
> 2. Discuss univalence and how we get "isomorphic structures are equal".
>
> 3. I will advertise Brent Yorgey's PhD thesis about combinatorial
> spieces, and probably cite some gems from it
> (
> https://homotopytypetheory.org/2016/07/20/combinatorial-species-and-finite-sets-in-hott/
> )
>
> I don't have a good feeling for what might pique a combinatorialist's
> interest. Does anyone here?
>
> With kind regards,
>
> Andrej
>
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Thread overview: 2+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-07-01 11:35 Andrej Bauer
2019-07-01 11:52 ` Gabriel Scherer [this message]

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Discussion of Homotopy Type Theory and Univalent Foundations

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