Category Theory
Zulip Server
Archive

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.

Stream: learning: questions

Topic: subcategory of the structure maps in a monoidal category

Nathaniel Virgo (Jun 20 2025 at 17:30):

Given a symmetric monoidal category $C$, I'd like to consider something like "the subcategory of $C$ consisting only of those morphisms that can be built up from the associator, unitors and symmetry".

I think that description doesn't really make sense on the nose (unless I'm wrong and it actually does?), so I'm looking for something that could be said to be "morally" that, if it exists.

A first attempt would be the category where objects are finite lists of objects in $C$ and the morphisms are permutations and insertions/deletions of the unit object. But this category doesn't know that the objects $[A,B]$ and $[A\otimes B]$ are related in any way, or $[A\otimes(B\otimes C)]$ and $[(A\otimes B)\otimes C]$. We could try adding extra morphisms to fix that, but I couldn't see a way to do that that would make the latter two objects isomorphic without making too many other things isomorphic as well.

The question is a bit vague because I don't know precisely what I want, but I figure if what I want does exist then someone will know it anyway.

Kevin Carlson (Jun 20 2025 at 17:31):

Why shouldn't it make sense? You can generate a subcategory from any class of maps you like.

Nathaniel Virgo (Jun 20 2025 at 17:32):

Hmm, good point. I guess what I meant is "I suspect it doesn't do what I want it to do", but then maybe I do need to be more precise about what that is. (I will attempt to do that, hang on)

Nathaniel Virgo (Jun 20 2025 at 17:47):

Here's what I'm trying to do, I realise it's a bit weird and specific. If $C$ is just a set then we can form the groupoid whose objects are finite lists of elements of $C$, where morphisms are ways of permuting the elements of one list to produce another.

But $C$ is actually a symmetric monoidal category, so I'm trying to categorify that into a double category. What I want is for the tight maps to be something like finite lists of objects of $C$ with morphisms as above, the loose maps to just be morphisms of $C$ (from the elements of one list all tensored together to the elements of another list all tensored together), and the squares to be commuting squares in the reasonably obvious sense involving the symmetry. This exists and it's fine, but it's a bit weird that the tight category doesn't really know anything about the monoidal product, so I was trying to fix that.

I could just let the tight category be the subcategory of $C$ consisting of only the structure maps, but my intuition is that that wouldn't really behave much like finite list of objects of $C$ any more, because an object might be expressible as a tensor product in many different ways and it wouldn't really respect that.

Kevin Carlson (Jun 20 2025 at 18:37):

It sounds to me a bit like you want the tight category to be the category of structure maps in the strictification of $C$, which does involve building lists of objects like this.

Kevin Carlson (Jun 20 2025 at 18:37):

And passing to the strictification, you avoid any coincidences of tensor products on objects.

Mike Shulman (Jun 20 2025 at 18:51):

What actual goal are you trying to achieve with this construction?

Nathaniel Virgo (Jun 20 2025 at 19:32):

I'm ultimately trying to define something like a double operad, but where it behaves like a symmetric monoidal category in one direction and a symmetric coloured operad in the other. I don't know if that'll work or not - I'm mostly just playing around with symmetric operad-like things to try and get a better feel for them.