Discussion of Homotopy Type Theory and Univalent Foundations
 help / color / Atom feed
From: Noah Snyder <nsnyder@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: [HoTT] Re: Dependent path composition in ordinary higher category theory
Date: Fri, 28 Feb 2020 10:44:18 -0800
Message-ID: <CAO0tDg5mstbw3Ujjfu+kMfgW7_7y7raxbWV_yh-2phB2ynwb8g@mail.gmail.com> (raw)
In-Reply-To: <CAO0tDg7uSiQrU8+0tnwJQHB_AWvRKAxmAePF88GOvd_ra=rpwA@mail.gmail.com>

[-- Attachment #1: Type: text/plain, Size: 2211 bytes --]

Mike pointed out that I didn't explain how my example is a special case of
dependent path composition.  A type family over S^1 is a higher groupoid
together with an (invertible) endofunctor F.  A path over loop is an
algebra for that endofunctor (where the map is an iso).  If you dependent
path compose it with itself n times you get a path over loop^n, i.e. an
algebra for F^n.  Best,

Noah

On Fri, Feb 28, 2020 at 10:33 AM Noah Snyder <nsnyder@gmail.com> wrote:

> Section 2.3 of the book introduces "dependent paths" (which are paths in a
> fibration "lying over" a path in the base) and "dependent path composition"
> which composes such dependent paths when that makes sense.  I'm working on
> a paper that's not about HoTT but where "dependent path composition" plays
> an important role.  The problem I'm running into is that I don't know what
> dependent path composition is called in "standard" mathematics.  Does
> anyone know if this has another name in higher category theory?
>  (Naturally, we'll include a remark mentioning the HoTT way of thinking
> about this (since it's how I think about it!), but I think that won't be
> illuminating to most of our target audience.)
>
> The simplest example of what I have in mind here is if C is a category and
> F is an endofunctor and c is an F-algebra (i.e. we have endowed c with a
> chosen map f: F(c) --> c) then c is also an F^n-algebra by taking the "nth
> power of f" which actually means f \circ F(f) \circ F^2(f) \circ ... \circ
> F^{n-1}(f).  In particular, I think this example illustrates that you can
> talk about "dependent k-morphisms" and their compositions without requiring
> anything in sight to be a (higher) groupoid.
>
> My best guess is that the right setting might be "indexed (higher)
> categories"?
>
> Best,
>
> Noah
>

-- 
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/CAO0tDg5mstbw3Ujjfu%2BkMfgW7_7y7raxbWV_yh-2phB2ynwb8g%40mail.gmail.com.

[-- Attachment #2: Type: text/html, Size: 3040 bytes --]

  reply index

Thread overview: 3+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2020-02-28 18:33 [HoTT] " Noah Snyder
2020-02-28 18:44 ` Noah Snyder [this message]
2020-03-02  7:03   ` [HoTT] " Michael Shulman

Reply instructions:

You may reply publicly to this message via plain-text email
using any one of the following methods:

* Save the following mbox file, import it into your mail client,
  and reply-to-all from there: mbox

  Avoid top-posting and favor interleaved quoting:
  https://en.wikipedia.org/wiki/Posting_style#Interleaved_style

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to=CAO0tDg5mstbw3Ujjfu+kMfgW7_7y7raxbWV_yh-2phB2ynwb8g@mail.gmail.com \
    --to=nsnyder@gmail.com \
    --cc=HomotopyTypeTheory@googlegroups.com \
    /path/to/YOUR_REPLY

  https://kernel.org/pub/software/scm/git/docs/git-send-email.html

* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link

Discussion of Homotopy Type Theory and Univalent Foundations

Archives are clonable: git clone --mirror http://inbox.vuxu.org/hott

Example config snippet for mirrors

Newsgroup available over NNTP:
	nntp://inbox.vuxu.org/vuxu.archive.hott


AGPL code for this site: git clone https://public-inbox.org/public-inbox.git