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Suraaj K S (Jun 05 2024 at 15:00):I was wondering if there was a connection between these three notions in CT?
Apart from using 'monoid' in all three definitions, is there another connection? For instance, can one be defined in terms of another?
Maybe something like a Monoid in Cat is a Monoidal category, etc.?
Spencer Breiner (Jun 05 2024 at 15:10):Almost. A strict monoidal category is a monoid in Cat, but a general monoidal category is only a [[pseudomonoid]].
Dylan Braithwaite (Jun 05 2024 at 15:11):Suraaj K S said:
Monoid is a category - A one object category is a monoid.
Also a one-obect bicategory is a monoidal category.
Suraaj K S (Jun 05 2024 at 15:16):Ah I see. Can the idea of 'monoids in a category' be derived from other notions?
Spencer Breiner (Jun 05 2024 at 15:30):Almost everything in category theory can be derived from something else, but if you walk down that path, you are apt to end up back where you started (or maybe one level up in an abstraction hierarchy).
At least one good representations for a monoid (in any monoidal category) is as a monoidal functor out of finite sets (with coproduct for the monoidal structure). It sends an -element set to an -fold tensor of the underlying object, and the unique arrows and to the multiplication and unit, respectively. All other functions can be built from these via coproduct and composition.
John Baez (Jun 05 2024 at 15:45):Spencer Breiner said:
Almost everything in category theory can be derived from something else, but if you walk down that path, you are apt to end up back where you started (or maybe one level up in an abstraction hierarchy).
True. But as Spencer knows, that is not a bad thing. At very least it's fun and educational - sort of like hiking along different trails to get to know the woods.
John Baez (Jun 05 2024 at 15:49):Suraaj K S said:
Ah I see. Can the idea of 'monoids in a category' be derived from other notions?
'Monoid in a category' is just barely meaningful, but 'monoid in a monoidal category M' is a very important concept, and there are several nice ways to define it. For example, it's a lax monoidal functor from the terminal monoidal category to M.
Vincent Moreau (Jun 05 2024 at 17:44):Spencer Breiner said:
Almost everything in category theory can be derived from something else, but if you walk down that path, you are apt to end up back where you started (or maybe one level up in an abstraction hierarchy).
At least one good representations for a monoid (in any monoidal category) is as a monoidal functor out of finite sets (with coproduct for the monoidal structure). It sends an -element set to an -fold tensor of the underlying object, and the unique arrows and to the multiplication and unit, respectively. All other functions can be built from these via coproduct and composition.
I may misunderstand your message, but I'm pretty sure that the property you describe relies instead on the augmented simplex category, i.e. the category of finite ordinals and order-preserving functions. I think that the category of finite sets would correspond to the case of commutative monoids.
Spencer Breiner (Jun 06 2024 at 01:54):Thanks, Vincent Moreau. I was indeed thinking about commutative monoids