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Max New (Feb 22 2022 at 13:17):So fc-multicategories/virtual double categories are a "nice place to define generalized categories" in the same way that monoidal categories are a "nice place to define generalized monoids". To define commutative monoids we need a symmetric monoidal category, (or possibly merely a braided monoidal category). Is there a published/folklore definition of what the generalization to a braided/symmetric fc-multicategory/virtual double category should be or do I need to just work out the details myself? Seems like this would be a "nice place to define generalized groupoids"
Max New (Feb 22 2022 at 13:19):Two guesses would be it's the same as generalized multicategories with respect to the free groupoid monad or free dagger category monad
Morgan Rogers (he/him) (Feb 22 2022 at 18:02):Huh? Why should the generalization of commutative monoids give you groupoids of any kind?
Mike Shulman (Feb 22 2022 at 18:22):Are you thinking of a vdc with a "reversal" operation on 1-cells? I wouldn't call that "braided" or "symmetric", and I don't know of a definition.
Max New (Feb 22 2022 at 23:23):Oh I guess my analogy with monoids doesn't quite make sense. But yes I'm thinking of a vdc where every "pro-arrow" can be reversed
Mike Shulman (Feb 22 2022 at 23:27):The free -category monad seems like a good idea to start with.
Max New (Feb 22 2022 at 23:55):Ah and maybe it's analogous to a star autonomous category rather than a symmetric monoidal category
Max New (Feb 22 2022 at 23:56):Or compact closed. I'll have to think about it