@SaalHardali No, Bousfield shows that Cobar(*,ΣX,*) is the Z-completion of X in the sense of Bousfield–Kan (at least if X is connected), so X → Cobar is a homology equivalence if and only if X is Z-good. Probably X → Cobar is always a homology pro-isomorphism when we regard Cobar as a pro-object, as happens for the Bousfield–Kan tower, but I'm not sure.

The standard examples of Z-bad spaces are the real projective plane and infinite wedges of circles.

In fact S^1 ∨ S^1 is already Z-bad!

A related question to the above: what's the state of the art regarding "derived completion" with respect to maps of connective E_n-ring spectra A-->B? I recall that Carlsson wrote something saying that if this map was an isomorphism in pi_0, (and maybe surjective on pi_1?) and A and B were commutative, that the limit of the Amitsur complex was equivalent to A. Can this be expected to hold more generally?

I'm particularly interested in references, rather than just "yes, that's true."

What's a good example to give my students of a Serre spectral sequence computation with non-constant coefficients?

@OmarAntolín-Camarena The orientation double cover of a manifold (say, of the projective plane to be concrete). The fiber is two points, and it is easy for students to see that the monodromy exchanges those two points.

@AndyPutman That's a good example of a local system (and one I already showed them), but the cover doesn't make for much of a Serre spectral sequence because the fibre is zero-dimensional. I mean, I should them that anyway, as a kind of degenerate example.

Hi guys, I have a quick question about the homotopy category. Let $\mathcal{A}$ be any abelian category and let $A^{\bullet}, B^{\bullet}, C^{\bullet}$ be any three complexes. Then do we have a Tensor-Hom adjunction $Hom(Tot(A^{\bullet} \otimes B^{\bullet}), C^{\bullet}) = Hom(A^{\bullet}, Hom^{\bullet}(B^{\bullet} \otimes C^{\bullet}))$?

@OmarAntolín-Camarena How about $\text{Pin}(2) \to SU(2) \to \Bbb{RP}^2$? The hard part here is identifying the action on the fiber (but really, you know what it has to be to get the right answer). You don't even have to calculate a local system's homology, since you can just apply unoriented Poincare duality!

Maybe the klein bottle then?@OmarAntolín-Camarena

(as a circle bundle over a circle). The spectral sequence is still kinda boring.

@fpqc Impostor!

Does anyone have a pdf of "Fibred and Cofibred Categories" by John Gray?

@OmarAntolín-Camarena Actually I think a better answer is to observe that applying Leray-Serre to a fiber sequence of K(G,1)s produces the Lyndon-Hochschild-Serre spectral sequence. The coefficient systems $H^*(N;A)$ are often/usually interestingly nontrivial, even if the underlying $G$-module $A$ has trivial action.

I haven't put any thought into this but it seems like you could get something nice for the usual nonsplit extensions of cyclic groups.

@JonathanBeardsley Let A-->B be a map of connective E_infinity-ring spectra. If A and B are noetherian and A has bounded above homotopy groups, then the totalization of the cosimplicial diagram Mod_{B \otimes_A ... \otimes_A B} is always the stable infinity-category of quasi-coherent sheaves on the formal completion of SpecA in the image of SpecB, in the sense of Lurie's book "Spectral Algebraic Geometry."

When Spec pi_0(B)-->Spec pi_0(A) is finite and surjective then this totalization will actually compute the module category Mod_A. That's what happens for example when pi_0(A)-->pi_0(B) is just an isomorphism as you wrote.

I think this is what Carlsson proved but the reference I know is Lurie's book, 5.6.6. He doesn't give the formal completion statement there but I'm pretty sure that Carlsson does give some version of it. If you're looking for an E_n-reference I strongly doubt it exists, but presumably it would follow if you repeat Lurie's proofs in E_n-spectral algebraic geometry and assume that all sufficiently natural statements about E_oo-geometry also hold in E_n-geometry (which is not quite the case...).

Does anyone know a good reference for an axiomatic treatment of the conjectural triangulated category of mixed motives and the motivic t- structure (together with all the weight structures etc...). Even better if it explains how the standard conjectures on algebraic cycles imply the existence of everything.

@SaalHardali The triangulated category of mixed motives is not conjectural: it's been constructed by Voevodsky (if I understand what you mean), what's conjectural is the abelian category of mixed motives (or relatedly, the motivic t-structure)

(that said I'd love to read a good survey about these topics)

(also, @Jon sorry I didn't see your emails, I'm going to try to answer them today)

@MikeMiller I agree, and I think Q_8 is a good example

@MikeMiller That's a good example, thanks.

@ThomasRot I think this is perfect, I'll definitely do this one. (That the spectral sequence is tiny just means it shouldn't the only example. :))

@MikeMiller Yeah, this the source of examples I had in mind. I'm planning on doing S_3 and Q_8.

@DenisNardin What I meant by "conjectural" is the ultimate dream of all properties and structures on this category that would exist given the standard conjectures.

@OmarAntolín-Camarena how about $C_2 \times C_2 \to D_8 \to C_2$? Also, I just sent you a document with a bunch of fun spectral sequence exercises

@DylanWilson Are you willing send me that document as well? :) My email is in my profile.

Here, use at your own risk (I haven't checked for errors or for accidentally giving problems that require a technique not listed): math.uchicago.edu/~dwilson/notes/spectral-sequence-practice.pdf

👌🗿

Power operations on the homotopy of E_\infty-algebras over height n Morava E-theory are controlled by a monad which Rezk calls T. Is it true/known that the free T-algebra on one generator is always polynomial as a ring? For n=1 this is a statement about free theta-algebras which is due to Bousfield. Seems like Yifei Zhu's calculations also imply it at n=2.

@PaulVanKoughnett this is the homotopy of the free E_infty-E-algebra on one generator, right?

@JonathanBeardsley one place where you can find stuff related to this q'n is prop'n d.3.2.1 and section d.3.3 of sag; as @Gasterbiter said, you can use proper descent in the connective E_oo-world (which in the case of maps of affines is descent for a finite morphism). presumably the proof provided in 5.6.6 of sag could also work for E_n-algebras?

@skd Yes. I have an idea of how to prove it, but it's also possible that it follows from some statement in the "rings of power operations are Koszul" paper I don't understand, or something.

If memory serves, that the completed E-homology of \bigvee_k B \Sigma_k is a completed polynomial ring is due to Kashiwabara; I think another proof also appears in Strickland's paper on the Morava E-theory of symmetric groups.

@WilliamBalderrama Great, thanks!

Right, yes, that polynomial-ness is important to strickland's proof, though I think he only credits Kashiwabara for the corresponding fact in Morava K-theory

one question i'm idly wondering about is the following. Spec E^0(BS_p)/tr is supposed to be the moduli of subgps of order p in the associated universal deformation fg G. this can be thought of (roughly) as the moduli of p unordered points in G which sum to zero

but BS_p is the moduli space of p unordered points in R^oo; there's no way that this is a coincidence. so is it possible to use this interpretation of BS_p to directly argue that Spec E^0(BS_p)/tr is what it is?

maybe this is already implicit in strickland's proof, in which case it'd be great to get a reference to some statement in his papers which talk about this

Strickland has some discussion of the relation between the symmetric group thing and the space of divisors (and subgroups) at the beginning of section 9 of his paper. It's not quite what you asked about, but I think you might find it interesting

@skd I think the right way to think about it is as follows -- sorry if I'm saying things you already know. E^0(BC_p) parametrizes p-torsion points in the formal group of E. S_{p-1} acts on this by moving a p-torsion point around the subgroup it generates, so E^0(BS_p) = E^0(BC_p)^{S_{p-1}} represents subgroups generated by p-torsion points. Then the transfer map from E^0 hits the trivial subgroup.

@DylanWilson Thanks Dylan!

@PeterNelson thanks! i'll check it out once i get the time. @PaulVanKoughnett thanks. i think i was wondering about a related but somewhat different q'n: can one directly obtain the "configuration space" view of Spec E^0(BS_p) using the configuration space model of BS_p?

this is probably in the section of strickland's paper that peter mentioned, so i should probably read that before asking any potentially silly questions that strickland might've already answered for me

For $\infty$-category fans: the preprint by Stevenson that appeared today looks very nice. arxiv.org/abs/1810.05233