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