is $\pi_\ast (E\widehat{\wedge} E \widehat{\wedge} \cdots \widehat{\wedge} E)$ concentrated in even degrees?

@JuanSebastianLozano this is a rather broad question, but if you want a nice introduction to like vector bundles and topological K-theory and things of that sort, you should look at Hatcher's book on this. there's also switzer's algebraic topology

whoops you said paper or survey article

@skd I've looked at some of the more elementary stuff in Hatcher, but I don't know exactly what it covers later on, but yeah, I asked for a paper or survey article because those tend to be more focused and give some more specific motivation.

@JuanSebastianLozano See the answer by Dylan here: math.stackexchange.com/questions/415787/…

note: it's about stable homotopy theory

@Saal For a derivator version of the 6-functor formalism see this:

home.mathematik.uni-freiburg.de/hoermann/h.pdf

@skd What is $\widehat{\wedge}$? And I suppose $E$ is even? Does the dual Steenrod algebra count as a counterexample?

@SeanTilson sorry, i should've specified that

np

that's the completed smash product (so like $L_{K(n)}(-\wedge -)$)

and $E$ is even periodic (more specifically, i'm thinking of E-theory at height n)

Ok.

That is a very interesting question. I guess I am not sure I would go about computing such a thing. What is the standard method? One of the only methods I know of for computing $K(n)$ -localization is via some inverse limit tower with generalized Moore spectra. That doesn't seem to play well with the ways I know how to compute smash products.

ok so the first nontrivial case is $E^\vee_\ast E$, right

what is this?

I think Rezk, and Barthel-Frankland might be a reasonable reference.

I don't know off the top of my head.

which paper of Rezk?

ok there are multiple

thanks @SeanTilson

Maybe arxiv.org/pdf/1204.4831.pdf and arxiv.org/abs/0902.2499

After definition 11.10 in the notes on blumberg's course there is mention of an "equivariant triangulated structure" on the genuine equivariant stable homotoppy category. Apparently this is unwritten but "the idea is that cofibre sequences induce long exact sequence of equivariant homotopy groups, where the shift is any choice S^V". Does anyone know more about this?

@TomBachmann if I recall correctly, the idea is that there's one kind of shift in a triangulated category, but in Ho(Sp^G), every representation V gives you a shift, namely \Sigma^V, and these satisfy some compatibility conditions

Can you give an example?

so you could write down the axioms of a "triangulated category with RO(G) shifts" and show Ho(Sp^G) satisfies it, but it's not clear if that's useful at all

re: example, maybe G = Z/2? There we have two kinds of shifts, trivial and sign: X[1] = ΣX and X[sign] = Σ^sign X. You would have two kinds of distinguished triangles, X -> Y -> Z -> X[1] and X -> Y -> Z -> X[sign], and these would satisfy some compatibility axiom, e.g. X[sigma][1] = X[1][sigma], as well as analogues of the examples of a triangulated category

*analogues of the axioms, sorry

I haven't thought about the details, though, so I don't know exactly how that would go. I hope this is helpful; I worry that I'm just restating trivialities

But what does a distinguished triangle X -> Y -> Z -> X[sign] mean? That you get (for example) a long exact sequence with shifts along the sign representation? If so how do you produce such a sequence?

ah, I don't know enough equivariant homotopy theory to answer that question :/

Ok. Thanks for trying :)

@skd $E^\vee_*E$ is known and is concentrated in even degrees. See Hovey's paper 'Operations and co-operations in Morava E-theory'.

It's isomorphic to $Hom^c(\G_n,E_*)$, where $\G_n$ is the extended Morava stabilizier group

For the higher terms, you can compute that there is an isomorphism \pi_*(E \widehat{\wedge} E^n) = E_*E \btimes ... \btimes E_*E, where \btimes is the L-complete (or in this case, I think, m-adically completed) tensor product. This is in my paper with Barthel, arxiv.org/pdf/1410.5269.pdf, Cor. 1.24 (it's probably elsewhere in the literature as well), and in particular is also concentrated in even degrees

(You can probably also deduce everything from various results in Hovey-Strickland's memoir on Morava K-theories)

@TomBachmann I think you just model the cofiber using a different model of the interval, so it wouldn't have the trivial actions. This might fall apart, and it certainly seems specific to $C_2$, but it may also be the case that the collection of these new distinguished triangles doesn't necessarily have to be very large, does it?

how do you extend to an action on the mapping cone if you put a nontrivial action on the interval?

It was a guess.

Doesn't the mapping cylinder then have a different C_2-action? or am I being stupid and it can't even be formed?

I guess your point is that the pushout doesn't make sense because... of the point having the trivial C_2 action.

Or maybe you have another idea about how to get these other triangles.

to put a C_2-action on the mapping cylinder, you need to have an isomorphism between the two ends, right?

to my knowledge, these kinds of distinguished triangles with equivariant shifts don't exist

what you certainly have is an action of the equivariant Picard group on ordinary distinguished triangles

including representation spheres

I think cofibre correspond to the usual one, but we can smash everything with representation spheres

So when we are looking at any spectral sequences along this line, we are automatically in RO(G)-grading, sometimes we even have nice RO(G)-graded multiplicative structure

@MingcongZeng But that doesn't give a triangle involving X which seems to be what the question was about.

Or are you both just saying that one should keep track of the action of the picard group?

My question was about the meaning of that comment. I agree that you can do ordinary cofibers and then smash with representation spheres at the end, but that does not sound (to me) like what was implied. I tried to think abot building an action into to cone interval but failed to see how.

Yeah, reading the comment and thinking for a while, it became more and more puzzling

I thought it too naively

I'd like to see the reference to this in Andrew's course

could someone post the link to the notes? I know it's in the starred messages pane somewhere but my browser window isn't big enough to see it and it doesn't scroll

ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf

here we go

yeah, I don't know what to make of that sentence

I mean, he says it hasn't been written down because it's "not super useful", but if you had an analog of a cofiber sequence in which the shift was by an arbitrary representation sphere, it would be super useful!

but this isn't even possible non-equivariantly, at least if you interpret it naively - what kind of cofiber sequence has a shift by a 5-sphere?

I mean you could do stuff using equivariant cubes, but I struggle to see those things as "sequences"

(Something like take a non-equivariant map X→Y where X is a G-space and use left Kan extend it to a functor from P(G_+) by putting X on the initial vertex and Y on all the non-basepoint vertices immediately connected to X and 0 on the basepoint)

Yes, using equivariant diagrams is the only thing which seemed vaguely like what was being implied, but i also struggle to see in what sense this is a "triangulated structure"

@Drew thanks!

is there an oo-categorical analogue of the Butz-Moerdijk result that a topos with enough points is the classifying topos of some topological groupoid?

@ArunDebray that'd be great! I think the subject is sorely lacking good notes, so your work is very appreciated.

@BrunoStonek Thank you, I will look at it!

Wow, that's a really great answer, it is definitely in the spirit of what I'm looking for.

can anyone recall where it was first proved that for a commutative ring (spectrum) A, we can compute $THH(A) = S^1 \odot A$ as the tensoring in CAlg(Sp)? or even HH(A), computed in sCRing or whatever

I believe that's McClure-Schwänzl-Vogt

sciencedirect.com/science/article/pii/S0022404997001187

ah yes, thanks

Suppose I have a square of simplicial spaces. I can take (fat) geometric realization and get a square of spaces. What sort of conditions do I have to check to ensure that this resulting square is homotopy cartesian?

@ChrisSchommer-Pries, is this assuming the original square of simplicial spaces was cartesian?

I have the following question, hopefully this is true and written down somewhere: Say a morphism $c \rightarrow c^\prime$ in an $\infty$-category C is an epimorphism if for any other object $d \in C$ the induced map on mapping spaces $map(c^\prime, d) \rightarrow map(c, d)$ is an inclusion of a union of path components (I believe Lurie calls such maps monomorphisms, but I might be wrong).

Say I have another $\infty$-category $D$, functors $X, Y: C \rightarrow D$ and a natural transformation $f: X \rightarrow Y$.

I'm looking for the following statement: If for all $c \in C$, $f(c): X(c) \rightarrow Y(c)$ is an epimorphism, then so is $f$ itself (in the functor category). Moreover, for any other $Z: C \rightarrow D$, a natural transformation $X \rightarrow Z$ factors through $Y$ if and only if for all $c \in C$, $X(c) \rightarrow Z(c)$ factors through $Y(c)$.

I've been struggling with this for some time now, although phrased like that it seems plausible

I would be happy with any characterization of monomorphisms/epimorphisms in a functor category

@ChrisSchommer-Pries There is the pi-star Kan condition of Bousfield and Friedlander, and if you don't like it there is math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf

A sufficient condition is that the degreewise \pi_0 of the lower right corner is a constant simplicial set, cf HA Lemma 5.5.6.17.

@PiotrPstrągowski Morally this should follow from the end formula for natural transformations (see Prop 2.3 in Saul's paper arxiv.org/pdf/1408.3065.pdf).

@ChrisSchommer-Pries there is a "model $\infty$-category" structure on sSpaces, and this model structure is right proper. so if you original square is a pullback, it suffices to check that one of the maps is a fibration, i.e. has RLP for all horn inclusions in the $\infty$-categorical sense. see arxiv.org/abs/1412.8411