[HoTT] Dependent path composition in ordinary higher category theory

[HoTT] Dependent path composition in ordinary higher category theory

[Noah Snyder]

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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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[HoTT] Re: Dependent path composition in ordinary higher category theory

[Noah Snyder]

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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,

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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Re: [HoTT] Re: Dependent path composition in ordinary higher category theory

[Michael Shulman]

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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