You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
David Egolf (Oct 29 2023 at 03:14):Thanks again to both of you for your comments!
I hope to read up a bit on monoidal preorders and enrichment, and then type up a more full response (hopefully in a few days!).
David Egolf (Oct 29 2023 at 21:53):I'm enjoying reading up on monoidal preorders and monoidal categories (in "Seven Sketches in Compositionality"). Currently, I'm trying to think of ways to get examples of monoidal categories.
I see on the nlab that "A monoidal category may be regarded as a bicategory with a single object". I'm wondering if it is also true that a bicategory with a single object always defines a monoidal category. If this is true, then because is a bicategory, then I think the endofunctors of a category and their natural transformations should form a monoidal category. Then I'd like to keep iterating this process, using the fact that the category of monoidal categories and strong monoidal functors forms a bicategory. I am hoping that a single object in this category together with its strong monoidal endofunctors, and the monoidal natural transformations between these forms a monoidal category. Then this process could be iterated, I hope.
If this does work as I'm hoping it does, then I think you can get in this way an infinite sequence of monoidal categories starting from an initial category. (I think it could be interesting to work out the case where we start with the category of the natural numbers viewed as a preorder, and so viewed as a thin category). (And I'm also wondering if all the automorphisms of an object in a bicategory together with the 2-morphisms between them form a monoidal category).
Looking at the definition of bicategory, if there is only a single object, the definition looks a lot like the definition for a monoidal category. And nlab notes "If there is exactly one 0-cell, say , then the definition is exactly the same as a monoidal structure on the category ". Does every object of a bicategory (together with its 1-endomorphisms and 2-morphisms between them) define a monoidal category?
Upon reflection, I would like to be able to prove (or disprove) this myself. I guess that's another reason to work some exercises around bicategories or monoidal categories!
John Baez (Oct 30 2023 at 09:15):David Egolf said:
I see on the nLab that "A monoidal category may be regarded as a bicategory with a single object". I'm wondering if it is also true that a bicategory with a single object always defines a monoidal category.
Yes: this is a case of the "periodic table" of n-categories, and it's one of the less problematic cases:
John Baez (Oct 30 2023 at 09:23):David Egolf said:
If this is true, then because is a bicategory, then I think the endofunctors of a category and their natural transformations should form a monoidal category.
That's true. And since is actually a 2-category, your proposed monoidal category is actually a strict monoidal category.
It's also easy to check directly that your proposed monoidal category is indeed a strict monoidal category.
A rather famous fact is that a monoid in this monoidal category is a monad. You've probably heard the joke: "a monad on is just a monoid in the monoidal category of endofunctors of ".
John Baez (Oct 30 2023 at 09:36):David Egolf said:
Then I'd like to keep iterating this process, using the fact that the category of monoidal categories and strong monoidal functors forms a bicategory. I am hoping that a single object in this category together with its strong monoidal endofunctors, and the monoidal natural transformations between these forms a monoidal category. Then this process could be iterated, I hope.
That's an interesting idea. It reminds me a lot of a slightly different idea, which is a way of getting from a monoidal category to a braided monoidal category. A monoidal category is a one-object bicategory, and is a tricategory, so gives an object in the tricategory .
Now, take , , all the 2-morphisms from to itself, and all the 3-morphisms between those. This is a tricategory with one object and one 1-morphism. So, it gives us a braided monoidal category!
This is called the center of the monoidal category .
John Baez (Oct 30 2023 at 09:39):You can get the center of a monoid (the submonoid of guys that commute with everything else) in a similar but less fancy way.
Jim Dolan and I came up with a super-general center construction that includes these two examples as special cases.
John Baez (Oct 30 2023 at 09:40):Taking the center marches us straight down the periodic table, increasing the amount of commutativity. But your variant construction does something else!
Notification Bot (Oct 30 2023 at 09:58):7 messages were moved here from #learning: questions > measuring similarity of objects with functors by John Baez.
John Baez (Oct 30 2023 at 09:59):David Egolf said:
Looking at the definition of bicategory, if there is only a single object, the definition looks a lot like the definition for a monoidal category. And nlab notes "If there is exactly one 0-cell, say , then the definition is exactly the same as a monoidal structure on the category ". Does every object of a bicategory (together with its 1-endomorphisms and 2-morphisms between them) define a monoidal category?
Upon reflection, I would like to be able to prove (or disprove) this myself.
I think you can do it. As you suggest, you mainly just need to compare the definitions of bicategory and monoidal category.
David Egolf (Oct 30 2023 at 23:59):Very cool! Thanks for explaining!
It's great that endofunctors of a category and their natural transformations form a (strict) monoidal category. That will give me lots of examples of monoidal categories to have fun with!
Mike Shulman (Oct 31 2023 at 00:15):By the way, your iterative process can be decategorified: starting with a set (or an object of any category, really), its endomorphisms form a monoid. Then you can look at the endomorphisms of that in the category of monoids, etc.