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Matteo Capucci (he/him) (Jan 28 2024 at 11:02):(I answered myself while writing this, still posting for posterity (i.e. likely me in a few weeks))
In this paper, Kock defines a morphism of pseudoalgebras for a KZ-monad as a map between the carriers such that:
image.png
His monads are endofunctors of a 2-category, and thus he can appeal to naturality of the unit to justify the existence of the square whose mate is taken. Now I'm working in a Gray-category and I have a pseudomonad there. How can I justify appealing to naturality of the unit anyway?
Answer: because I'm actually appealing to naturality of whiskering by the unit.
Notification Bot (Jan 28 2024 at 11:02):Matteo Capucci (he/him) has marked this topic as resolved.
Matteo Capucci (he/him) (Jan 28 2024 at 11:03):Maybe someone can clarify a further point: is whiskering still strictly natural or just pseudonatural? I suspect the latter for both Gray and tricategories
Patrick Nicodemus (Jan 29 2024 at 13:59):Matteo Capucci (he/him) said:
Maybe someone can clarify a further point: is whiskering still strictly natural or just pseudonatural? I suspect the latter for both Gray and tricategories
Whiskering should define a strict functor between bicategories of pseudofunctors. Is that what you mean?
Matteo Capucci (he/him) (Jan 29 2024 at 13:59):Kinda: I'm working in a general tricategory, not necessarily in
Patrick Nicodemus (Jan 29 2024 at 14:03):Matteo Capucci (he/him) said:
Kinda: I'm working in a general tricategory, not necessarily in
Ok. Yeah I have no idea. To elaborate a bit, I'm pretty sure that if are bicategories, a pseudofunctor, then you have a strict pseudofunctor , but this appeals to a trick where you use the pseudofunctor directly and don't mention the unit pseudonatural transformation on . In a general tricategory I presume you usually would define only the composition bifunctor and then whiskering is the special case where one of them is taken to be the unit 2-cell, so the trick would not work, and my comment does not apply.
Patrick Nicodemus (Jan 29 2024 at 14:05):But it would be interesting to talk about tricategories that supported a notion of "true whiskering" where a 2-cell is actually whiskered with a 1-cell rather than a unit 2-cell, and this was somehow a different operation than just composition of 2-cells.
Patrick Nicodemus (Jan 29 2024 at 14:12):By the way have you ever seen a definition of a KZ monad on a tricategory?
Matteo Capucci (he/him) (Jan 29 2024 at 16:25):No :/