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Shea Levy (Nov 19 2020 at 20:05):Do and have the same relationship as and ? If it matters, mostly interested in the 2-category of weak monoidal categories
Nathanael Arkor (Nov 19 2020 at 20:12):Arguably Set : Mon :: Cat : StrMonCat, since both Mon and StrMonCat arise as categories of internal monoids, whereas MonCat is a category of pseudomonoids.
James Wood (Nov 19 2020 at 20:14):When you think of sets as models of the trivial algebraic structure, then the original analogy sounds right. Monoidal categories are categories with functors and natural isomorphisms doing monoidy stuff; categories don't have any of that sort of structure.
Chad Nester (Nov 19 2020 at 20:21):We can make what James said a bit more precise: A monoidal category is an internal category in , while a (small) category is an internal category in .
Shea Levy (Nov 19 2020 at 20:22):Thanks!
John Baez (Nov 19 2020 at 22:49):We can be a bit more precise: a *strict* monoidal category is an internal category in Mon. This is a monoidal category where the associators and unitors are identity natural transformations.
John Baez (Nov 19 2020 at 22:49):A strict monoidal category is also a monoid internal to Cat.
John Baez (Nov 19 2020 at 22:51):An ordinary monoidal category is not usually an internal category in Mon.
John Baez (Nov 19 2020 at 22:52):I'm agreeing with @Nathanael Arkor here.
Mike Stay (Nov 20 2020 at 18:24):If we start with the 2-theory of pseudomonoids rather than the 1-theory of monoids, all interpretations of the 2-theory in Set (thought of as a 2-category) are strict monoids because all the 2-morphisms in Set happen to be identities. But the interpretations of the theory in Cat are arbitrary monoidal categories, so from that point of view, I'd say the original analogy is accurate.
John Baez (Nov 20 2020 at 20:26):Yes, the analogy is "morally correct", and you can make it technically correct using pseudomonoids.