Corollary 2.4.4. (use of naturality and whiskering) about book

[mikeshulman]

It's true in general that the book could stand more exposition of the category-theoretic terminology that it uses, but that's a big enough change that I doubt it will happen. I do think (unsurprisingly, since I'm a category theorist) that there's value in including the terminology -- in your case, it motivated you to (re)learn it! But perhaps we could add a parenthetical here noting that "naturality of H" is nothing other than the just-proved Lemma 2.4.3.

[mikeshulman]

I'm unsure why we need to whisker by (Hx)−1. Does (Hx)−1 not exist by Lemma 2.1.1? Then we can simply apply the fact that if p=q where p,q:x=y and r:y=z then p⋅r=q⋅r together with the equation from the first part of the proof and all the neccesary paths from Lemma 2.1.4. to arrive at the desired identity.

This is precisely a construction of whiskering. Theorem 2.1.6 defines it by direct path-induction on r, but you can also construct it out of other operations in the way you describe. (see below)

[IanRay11]

I like the use of the categorical language as long as it parallels the necessary and self contained information that comes before. I agree use of naturality should be relegated to a note or parenthesis (as you mentioned). A few paragraphs prior it is mentioned that a homotopy may be regarded as a natural isomorphism but no further context is given. For this reason I really do feel it is not sufficiently clear language for a proof.

[IanRay11]

So if I instead whiskered by $(Hx)^{-1}$ would I be doing path induction on... $(Hx)^{-1}$?

[mikeshulman]

The operation of "whiskering by r" can be defined for all r at once by doing path induction on r, or it can be defined for any particular r by the operations you described.

[mikeshulman]

Sorry, my brain must have been turned off. The correct answer is that this:

the fact that if p=q where p,q:x=y and r:y=z then p⋅r=q⋅r

is whiskering.

[IanRay11]

I think I was just starting to realize this! Thank you for clarifying!

[IanRay11]

Whiskering is admittedly a very cool name, but is it not a very desirable property of how identity and path composition should play together?

For any set with binary operation $(X,\cdot)$ we would hope that if $x = y$ then $x \cdot z = y \cdot z$. (Well-defined maybe?)

Is this an example of some of the "coherence laws" that were mentioned following Lemma 2.1.4. and even the Lemma and Corollary in question?

[mikeshulman]

It's true, it's also definable as $\mathsf{ap}_{\lambda u. u\cdot z}(p)$ for $p:x=y$.