Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Noah Snyder <nsnyder@gmail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Re: Dependent path composition in ordinary higher category theory
Date: Sun, 1 Mar 2020 23:03:26 -0800
Message-ID: <CAOvivQzAoLQkXUNDkauM5so1rNu9-LsyWRHM4xO16d4O8Wt-0A@mail.gmail.com> (raw)
In-Reply-To: <CAO0tDg5mstbw3Ujjfu+kMfgW7_7y7raxbWV_yh-2phB2ynwb8g@mail.gmail.com>

Yes, I think probably you want indexed categories, incarnated as
fibrations.  You can then remove the invertibility too: if C is the
1-object category corresponding to the monoid of natural numbers, then
an opfibration over C is a category equipped with an endofunctor, an
endomorphism over the generating morphism of C is an algebra for that
endofunctor, and its n-fold composition with itself is the
corresponding algebra for F^n.

On Fri, Feb 28, 2020 at 10:44 AM Noah Snyder <nsnyder@gmail.com> wrote:
>
> 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
>
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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 ` [HoTT] " Noah Snyder
2020-03-02  7:03   ` Michael Shulman [this message]

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