(resending, because categories@ only takes plain text; sorry for the duplicate, Mike; also asked at http://mathoverflow.net/questions/141022/naming-ambiguity-in-constructive-po...): In the HoTT Book, the choice is made to talk about whiskering, rather than horizontal composition, because horizontal composition is ambiguous and only defined up to paths. Naturally, there is left whiskering and right whiskering, and I think these are well-known to the category theory community. When working on defining natural transformations pointwise, there is a similar problem, and I would like to know if there are good names for the two variants. More precisely, given precategories C, C', D, C', functors F, G : C -> D, functors F', G' : C' -> D', and natural transformations T : F -> G and T' : F' -> G', and letting [A, B] denote the precategory of functors from A to B if A and B are precategories, and letting [F, G] denote the functor [C, D] -> [C', D'] if F is a functor C -> D and G is a functor C' -> D', we can form the pointwise natural transformations [F, F'] -> [G, F'] and [F, F'] -> [F, G']. Is there a good and/or standard name for these natural transformations? (Is one of them obviously "left" and the other "right"?) I'll probably end up notating the first as [T, F'] and the second as [F, T'], but I'd still like good names for them. On Monday, September 2, 2013, Michael Shulman wrote:
This is really a category theory question, not a homotopy type theory question, so you might have better luck on the categories mailing list, the nForum, or MathOverflow. The notations [T,F'] and [F,T'] are good, but I don't recall hearing a particular name for these before. ("pointwise natural transformation" certainly didn't call up the right image.) Maybe pre-whiskering and post-whiskering?
The reason I called them "pointwise", I think, was because I used them to construct the fact that if F ⊣ G, then [id, F] ⊣ [id, G], which I called pointwise adjunctions, perhaps due to some connection I thought they had with pointwise Kan extensions (but which I can no longer recall). Which one would be pre-whiskering and which one would be post-whiskering? Thanks, Jason [For admin and other information see: http://www.mta.ca/~cat-dist/ ]