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Matteo Capucci (he/him) (Feb 13 2022 at 18:42):Does anyone know where I can find a reference for the definitions in [[vertical transformation]]?
Matteo Capucci (he/him) (Feb 13 2022 at 18:44):I can't really understand if the definition there is really 'the' definition of natural transformation of double functors. They write: image.png
Matteo Capucci (he/him) (Feb 13 2022 at 18:46):But e.g. Parè defines natural transformations as follows:
Matteo Capucci (he/him) (Feb 13 2022 at 18:46):
Matteo Capucci (he/him) (Feb 13 2022 at 18:46):
Matteo Capucci (he/him) (Feb 13 2022 at 18:47):which makes quite a bit of sense since 'vertical' (or 'horizontal') transformations seem to ignore half of the structure of the functors
Mike Shulman (Feb 13 2022 at 19:37):Pare's definition is the same as a horizontal natural transformation.
Mike Shulman (Feb 13 2022 at 19:38):The nLab also gives an explicit version of the definition, which is just the transposed version of Pare's specialized to the case of strict functors.
Matteo Capucci (he/him) (Feb 13 2022 at 23:32):Uh, that's suprising, they look quite different to me (transposition aside)!
Matteo Capucci (he/him) (Feb 13 2022 at 23:34):Can you help me understand the correspondence between the two? Nevermind!
Matteo Capucci (he/him) (Feb 13 2022 at 23:35):I see the data and the first condition in Parè ('horizontal naturality') are the same
Matteo Capucci (he/him) (Feb 13 2022 at 23:38):Mmh ok 'vertical functoriality' is also the same but Parè's look funnier bc and are lax in his definition, so there's a bunch of lax constraints involved
Matteo Capucci (he/him) (Feb 13 2022 at 23:38):Ok I think I get it
Matteo Capucci (he/him) (Feb 13 2022 at 23:42):Is the definition completely transposable? Or does the following remark ruin the fun?
image.png
Mike Shulman (Feb 14 2022 at 00:38):For strict double functors it's completely transposable, and so the category of strict double categories and strict double functors is actually enriched over itself (indeed, it's cartesian closed). For lax double functors, which are lax in one direction but not the other, I think only the corresponding direction of transformation makes sense.
Matteo Capucci (he/him) (Feb 14 2022 at 16:58):I see, thanks!!