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Asad Saeeduddin (Sep 12 2021 at 06:56):Hello there. I was wondering if anyone knows of a high level map of "pseudoadjunctions" between Cat and various other 2-categories of more structured categories, such that there is a "free such and so category generated by a given this and that category" (where this and that may be "arbitrary", in which case we're talking about an adjunction out of Cat). I often end up wondering how to equip categories that may already have a certain given form of structure freely with "missing" structure, and knowing whether the relevant 2-categorical analogue of an adjunction exists (and how it works) would be very useful
James Deikun (Sep 12 2021 at 12:52):I don't really know of a 'map' for Cat but you can reuse a lot of your knowledge for Set. Any adjunction that comes from a planar operad has a pseudo version and you can probably tell when a categorical version is that thing, e.g., monoidal categories for monoids. I'm slightly less 100% on this, but I also think every symmetric operad has symmetric and maybe braided pseudo versions, e.g. commutative monoids -> symmetric monoidal categories, braided monoidal categories. And this should lift the whole map for the relevant sort of objects, not just the things connecting directly to Set.
John Baez (Sep 12 2021 at 18:08):It sounds like you may be entering the territory of the famous Blackwell-Kelly-Power paper.