Take the "strict polynomial ring" S[t] = \Sigma^\infty_+ (Natural numbers). I think the augmentation S[t] -> S sending t to 0 can be made an E-infinity ring map, so I can look at the indecomposables TAQ(S[t]) with respect to this augmentation. Is this something recognizable?

@WilliamBalderrama It's ⊕ Σ Sym^k(𝕊)/Sym^{k-1}(𝕊), the suspension of the associated graded of a filtration on Sym^∞(𝕊) = Hℤ. (It's also apparently closely related to the spectral Lie operad? But that's outside my expertise.) The absolute TAQ of 𝕊[t] is ⊕ Sym^k(𝕊); t=0 gives you this associated graded and t=1 gives you Sym^∞(𝕊) = Hℤ. Nick Kuhn studied this filtration a lot in the course of working on things related to the Whitehead conjecture.

@TylerLawson Oh that's neat, thanks. Is there any easy way to see this fact about the absolute TAQ of S[t]? (Or do I just need to read Nick's papers?)

@WilliamBalderrama I mostly know it because Sym: Spaces -> E_∞ Spaces and 𝕊[-]: E_∞-spaces -> E_∞ rings are both hocolim preserving, which lets you calculate the "iterated bar construction" on 𝕊[t] = 𝕊[Sym(*)] to be 𝕊[Sym(S^n)]; following the rest of the recipe for TAQ gets you the symmetric powers of spheres.

associative algebra : A_inf algebra :: Hopf algebra : _____

How does one fill in the blank? (I threw "H-infinity" at the wall and saw that I got power operations instead.) how about if you replace A_inf algebra by "A_inf category?" And is there a place where any such notion is written down which is more likely to be accessible to the simple-minded about these things, such as myself?

Then there are the various things you could demand like coassociativity and such, or whether or not you care about antipodes, ... it seems like a zoo. But at least somewhere to start would be nice.

@MikeMiller I think that a Hopf algebra is an associative coalgebra object in the category of associative algebras (with the coproduct monoidal structure). This is generalizable because we have E_1 algebras and E_1 coalgebras in a monoidal category.

@S.carmeli Wouldn't it rather be the tensor product monoidal structure ? (this is not the same for E_1-algebras)

@MaximeRamzi yes! sorry

I had E_infty in mind obviously

OK, I'll see if I can extract an explicit description of what this looks like in chain complexes at some point. Thanks.

@MikeMiller I think this is going to be a huge mess to try to suss out explicitly.

E.g. there won't be an operad for it.

But there's this definition at least: ncatlab.org/nlab/show/differential+graded+Hopf+algebra

I guess by "explicitly" I mean in terms of something like a list of operations, in the same way that we describe A_∞-algebras in chain complexes.

There's also the PROP of bialgebras BiAlg, and monoidal functors BiAlg→Ch(R) will be equivalent to bialgebras in Ch(R), so that gives a description of bialgebras at least which is somewhat explicit.

I guess there's also a Lawvere theory for Hopf-algebras

And a PROP

I'm aware of dg-Hopf algebras. I have a chain complex that I can cook up a bunch of operations on. Some of these operations form an $A_\infty$ structure, some of them go the other way. I'm not sure what kind of compatibilities I ought to be proving or expecting and so was hoping for something guiding.

It's good to know there's no operad in sight. I don't know what PROPs are but maybe that's what I want. I dunno. This is not the kind of setting where one expects to cook up an action of a fancy operad as opposed to a more operations-and-relations kind of construction.

So maybe the punchline is "there are plenty of interesting things to work on, move on".

Hi, I was reading a question where OP stars by mentioning the asymptotic $\sum_{n \in \mathbb{N} } \frac{\phi(n) } {n} w(n/x)=c_0(w) x+c_1(w) (\log^2 x ) +o(\log^2 x )\tag{*}$, I don't know much so I asked for a reference to see If I could understand further and he linked a paper mentioning equation number 9 which is:

$\sum_{n \in \mathbb{N} } \frac{n } {\phi(n)} (1-\frac{n}{x})=\frac{A}{2}(w) x-\frac12\log x+\frac{(1-D)}{2}+o(x^{-1/5} )\tag{}$, so I was wondering if someone could help me understand how does one go from () to (*) (as I understand $w(n/x)=(1-\frac{n}{x})$ )

$\sum_{n \in \mathbb{N} } \frac{n } {\phi(n)} (1-\frac{n}{x})=\frac{A}{2}(w) x-\frac12\log x+\frac{(1-D)}{2}+o(x^{-1/5} )\tag{°} $, so I was wondering if someone could help me understand how does one go from (*) to (°) (as I understand $w(n/x)=(1-\frac{n}{x})$ )

This is not related to homotopy theory, so this is not the right place to ask and you are unlikely to get an answer here.