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] Dependent path composition in ordinary higher category theory
Date: Fri, 28 Feb 2020 10:33:58 -0800
Message-ID: <CAO0tDg7uSiQrU8+0tnwJQHB_AWvRKAxmAePF88GOvd_ra=rpwA@mail.gmail.com> (raw)

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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)



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

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Discussion of Homotopy Type Theory and Univalent Foundations

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