I don't suppose anyone has access to this paper, and might be able to send me a copy? link.springer.com/chapter/10.1007/978-3-030-29597-4_5

I have "found" a copy

Oop nevermind me too.

I guess it's in some weird book published for Connes' birthday

Oh just got it, thanks so much Ian.

Could someone point to a reference on how to read the infinity page of an Adams-type spectral sequence? the different indexing to (co)homolgical s.s. confuses me a bit

Can an H-space be strictified (non-uniquely) to a topological group? Is this obvious? If not, is it obvious that it can be strictified to a topological monoid with inverses up to homotopy? Same question for co-H-spaces, which I'm more interested in, but I figure if I know how to answer one the other will be clear.

@Nikitas the columns are the (associated graded of the) homotopy groups of the answer, the vertical direction is the filtration

@MikeMiller no it can't: an H-space need not even be homotopy associative; there's an infinite hierarchy between an H-space (A_2-space) and a topological monoid (A_infty space)

Right, of course. Thanks.

but an A_infty space can be strictified to a topological monoid, fwiw

Right, $\Omega BA$. Thanks for pointing out where I started going the wrong way there.

(well $\Omega BA$ will only be the right thing if A is group-like, but yes!)

Let $\mathcal{C}^{\otimes}$ and $\mathcal{D}^{\otimes}$ be symmetric monoidal $\infty$-operads and let $\mathcal{C}$ and $\mathcal{D}$ denote the corresponding symmetric monoidal $\infty$-categories. Does every equivalence of $\infty$-operads $\mathcal{C}^{\otimes} \simeq \mathcal{D}^{\otimes}$ come from a symmetric monoidal equivalence $\mathcal{C} \simeq \mathcal{D}$?

In less precise terms: is every lax symmetric monoidal equivalence automatically (strict) symmetric monoidal?

@SaalHardali I think so: if $F$ is your lax symmetric monoidal equivalence, you need to verify if the maps $1→F1$ and $F(-)⊗F(-)→F(-⊗-)$ are equivalences. Let $G$ be its lax symmetric monoidal inverse. Then the composition $1→G1→FG1$ and $1→F1→FG1$ are equivalences,so $1→F1$ is an equivalence: in fact $1→G1$ has a left inverse ($G1→GF1$), but then $F1→FG1$ has a left inverse and it has also a right inverse ($1→F1$), therefore it is an equivalence.The other is fiddlier but I believe it is the same.

@DenisNardin @SaalHardali isn't the norm map of a pointed category a lax structure on the identity from C with coproducts to C with products?

@S.carmeli I'm taking a "lax equivalence" to mean something stronger than that: I'm assuming it is an equivalence in the cat of operads and lax monoidal functors, i.e. there exists $G:\mathcal{D}\to \mathcal{C}$ such that $FG\cong \mathrm{id}$ and $GF\cong\mathrm{id}$ as lax monoidal functors. Otherwise, as you say, the result is false

OK yah in that case your right

To say it differently, I'm proving that the subcategory of symmetric monoidal ∞-categories and symmetric monoidal functors is replete in the ∞-category of ∞-operads

@DenisNardin Nice argument! I hope the rest of the argument will not be difficult now that i've seen this. Thanks!

Nah, the other proof is pretty much the same, just with bigger formulas

You're welcome

@DenisNardin @SaalHardali I wonder if there's a trick with envelopes that makes it shorter? Like, Env(F) is now a symmetric monoidal equivalence, and we also have symmetric monoidal functors Env(C)-->C and Env(D)-->D which seem to make the square commute. Those are essentially surjective so I think one gets that the lax structure map is an equivalence since it was in the envelopes?

If X is a quasi-geometric E_2-stack, then is QCoh(X)^cn going to be Grothendieck prestable? In the E_oo - case this is SAG 9.1.3.1, but I don't know the SAG stuff well enough to verify this without spending an unreasonable amount of time backfilling.