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: 3- (and higher) cells for monoid objects?

Shea Levy (Oct 21 2020 at 18:28):

Continuing my quest to understand unbiased definitions, I've noticed we have:

• For any $n \in \mathbb{N}$, an arrow $\mu_n : M^n \to M$
• For any $l \in List(\mathbb{N})$, a 2-cell (possibly invertible) $assoc : \mu \circ (\mu \otimes ... \otimes \mu) \Rightarrow \mu \circ \gamma$, where $\gamma$ is the "flattening" arrow in the ambient unbiased monoidal category

It seems like we should be able to continue this, and get something like a sequence of 3-cells for any $(List \circ List \circ ... \circ List) \mathbb{N}$, where on one side you have the 2-cell mapping $assoc$ over and over into the nested list until you get to the base, and step by step you replace some of those with flattening the lists until on the right side of the final 3-cell you have by flattening the whole list.

Is this a known construct? It seems like you can generalize indefinitely forwards (and one step backward, starting with the object M itself), maybe you could stick a $\forall (n \in \mathbb{N})$ in front and get an $\omega$-cell (and then keep going?)