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: why are generators of 2-morphisms usually nullary?

Joshua Meyers (Mar 29 2021 at 02:07):

For $0$-morphisms (objects), we have nullary generators like $1$ and binary generators like $\otimes$. For $1$-morphisms, we have nullary generators like $\text{id}_c$ and $\alpha_{a,b,c}$ and binary generators like $\circ$ and $[,]$. For $2$-morphisms, we have a lot of $0$-generators like coherence laws, but no other arities. Why not?

Mike Shulman (Mar 29 2021 at 03:33):

Well, first of all, I think it's better to think of $\rm id$ as a unary operation that takes a 0-morphism to a 1-morphism. Similarly, $\alpha$ (by which I assume you mean the associator in a monoidal category) is a ternary operator, taking a triple of 0-morphisms to a 1-morphism. So a coherence law for a monoidal structure, regarded as a 2-morphism in a locally discrete 2-category, is not 0-ary but takes a bunch of inputs that are objects and morphisms.

Mike Shulman (Mar 29 2021 at 03:34):

As for why there are no operations taking 2-morphisms as inputs, that's just a function of the dimension of the category you're considering. In a monoidal 2-category, for instance, there are plenty of operations taking 2-morphisms as inputs.