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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