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Matteo Capucci (he/him) (Nov 26 2023 at 15:26):It also made me wonder something which might be right up @John Baez's alley. It seems we have the following progression:
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Matteo Capucci (he/him) (Nov 26 2023 at 15:27):However, sylleptic doesn't seem to appear, which is suspicious. Could I be missing something?
Matteo Capucci (he/him) (Nov 26 2023 at 15:28):The actual question is: if is a pseudocategory object in (resp. , ), does it make it automatically monoidal (resp. braided, symmetric)? And thus does sylleptic never appear?
Notification Bot (Nov 26 2023 at 15:29):3 messages were moved here from #learning: questions > Para construction and symmetric monoidal categories by Matteo Capucci (he/him).
John Baez (Nov 26 2023 at 19:16):I seem to remember some other case where the sequence skips near the end like that.... oh yeah, it's even worse: there's no difference between a monoidal transformation, a braided monoidal transformation and a symmetric monoidal transformation (between symmetric monoidal functors between symmetric monoidal categories).
Mike Shulman (Nov 26 2023 at 19:26):And similarly there's no difference between a braided monoidal functor and a symmetric monoidal functor (between symmetric monoidal categories). Those cases are because there are "no more dimensions available" at the top to add more structure or conditions -- is that what's happening with Para?
Mike Shulman (Nov 26 2023 at 19:28):Maybe an even closer analogy is that a monoidal fibrant double category gives a monoidal bicategory, a braided monoidal fibrant double category gives a braided monoidal bicategory, and a symmetric monoidal fibrant double category gives a symmetric monoidal bicategory -- there's no way to get a non-symmetric sylleptic monoidal bicategory out of a double category.
Matteo Capucci (he/him) (Nov 27 2023 at 07:36):I see, that's comforting!
Matteo Capucci (he/him) (Nov 27 2023 at 07:38):Mike Shulman said:
And similarly there's no difference between a braided monoidal functor and a symmetric monoidal functor (between symmetric monoidal categories). Those cases are because there are "no more dimensions available" at the top to add more structure or conditions -- is that what's happening with Para?
I'd say yes, specifically this is happening with actegories, whose monoidal structures don't seem to be able to induce something in-between braided and symmetric.