Discussion of Homotopy Type Theory and Univalent Foundations
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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	[thread overview]
Message-ID: <CAO0tDg5mstbw3Ujjfu+kMfgW7_7y7raxbWV_yh-2phB2ynwb8g@mail.gmail.com> (raw)
In-Reply-To: <CAO0tDg7uSiQrU8+0tnwJQHB_AWvRKAxmAePF88GOvd_ra=rpwA@mail.gmail.com>

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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,


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

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  reply	other threads:[~2020-02-28 18:44 UTC|newest]

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

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