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Hi Aaron,
I have looked at it. Interesting!
I am pretty sure that we don’t have “this” on the nLab and I can’t readily recognize anything really closely related that would be. But possibly somebody else here may have a deeper insight.
I always assumed that “the richest possible structure” refers to something rather simple. Namely, if F:𝒞→𝒟 is any functor, then you can consider the monoid EndF of endotransformations of F which acts on FX for every X∈𝒞 just by evaluating transformations at X. This monoid action encodes all natural transformations you can put on FX. Of course sometimes this monoid itself has more structure, for example if 𝒟 is the category of chain complexes then EndF is really a dg-algebra. In general it is a monoid in whatever monoidal category 𝒟 is enriched in. You may also want to consider operations of many variables in which case you will get an operad rather than just a monoid (I guess for this 𝒟 should be monoidal itself).
If you mean something more complicated than that could you give the simplest example that isn’t covered by what I suggested? I tried reading examples in your note, but I don’t quite follow them.
I’ve only taken a brief look at your notes, but it looks a lot like what Sarah Whitehouse and I are thinking about with Tall-Wraith monoids. See The Hunting of the Hopf Ring for details.
I was a little bit confused in the section on p-adic K-theory. You wrote K* but said “p-adic K-theory”. Do you mean cohomology or homology? And what is “MoravaMod”? With no explanation, I would expect that to mean modules over the coefficients of Morava K-theory (presumably with n=1 as you start in ordinary K-theory), but that is mod p, not p-adic. This section would appear to be referring to Bousfield’s work but you don’t mention it.
To use itex you need to use the Markdown+iTeX input formatter.
I still don’t see anything not covered by Tall-Wraith monoids, but as I said I haven’t looked through in any great detail.
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