@user40276 There's an explicit definition of a 4-category somewhere online (maybe by Trimble?), I don't think anyone's tried to write explicit diagrams for an n-category with n>4

@RuneHaugseng How does that relate to E_k algebras for k > 1? One can take automorphisms of the identity or something like this, but there doesn't seem to be straightfoward way to get the coherence diagrams for this E_k-algebra unless I'm overseeing something.

@user40276 As in, I don't think there's a definition of this sort for n-category for any n > 4, let alone for E_k-structures on them. But maybe you meant something different by "write down coherence diagrams" than the way I interpreted it (as "write out all the diagrams explicitly")

@RuneHaugseng Ah, ok. I mean writing the coherence diagrams explicitly or knowing a procedure in order to write them down explicitly. Now, I see that I got confused on the indexes. It should be k < n+2, k>2 and n> 2. Otherwise everything already get symmetric monoidal.

@user40276 As people have said, there's already a bottleneck trying to write down what a weak $n$-category is with explicit coherence diagrams. But within this constraint, there are cases of this which have been done. I think there are explicit definitions of $E_1$-monoidal $n$-categories (i.e. monoidal $n$-categories) for $n \leq 3$, $E_2$-monoidal $n$-categories (i.e. braided monoidal) for at least $n \leq 2$, and $E_3$-monoidal $n$-categories (so-called sylleptic monoidal) for $n \leq 2$.

The software globular also in some sense runs on a model of weak 4-categories with explicit coherences, and which in principle generalizes to all dimensions. But as I recall, this model has not been compared with other models

Do we actually know that all those pages of diagrams for a symmetric monoidal 2-category really correspond to E_infty 2-categories?

@TimCampion Thanks for the reply. I think that I would already be happy with the coherence diagrams in a E_3-monoidal 2-category. The other cases, I could write down myself, but for k > 2 I don't really know how to proceed or, at least, I'm not confident enough. Do you know a reference for that?

arxiv.org/abs/1301.1053 starting page 9

i'm trying to find a document: it's a book(?) mahowald and alan unell, "Lectures on Bott periodicity in stable and unstable homotopy at the prime 2". it doesn't seem to live on the searchable web. does anyone happen to have a copy?

@ReidBarton It seems that it only contains the diagrams for E-2-algebras unless I'm overseeing something. You also mentioned symmetric monoidal, but, in this case, you would need at least E_4...

Ah, ok. Forget that

Sylleptic=E_3 is on page 19, definition 4.7

@ReidBarton Thanks for the reference. I guess I will have a hard time in order to see that it really describes the structure of an E_3 and E_4 algebra (consequently, E_\infty).

Indeed

Does anyone know what the map in 1.16 of: arxiv.org/pdf/1703.07842.pdf is? $d_{K(1)}(R) := map(R, L_{K(1)}Pic(\mathbb{S}_p)$ and $j_{\mathbb{Z}_p} \in \pi_0d_{K(1)}(R)$, it seems like $\cdot j_{\mathbb{Z}_p}$ should be the adjoint of a map $(?, j_{\mathbb{Z}_p}): R \otimes R \to \omega_{K(1)}$ but i'm not sure what the map on the first factor should be

d_K(1)(R) is an R-module, so that map in 1.16 is just the extension of S^0 -> d_K(1)(R) detecting j to an R-module map R -> d_K(1)(R)

Ohh, @skd thank you!