Saul Glasman

If you started with a chain complex in an abelian category, then the spectral sequence you get out has to degenerate at E_1, and I'm guessing that means that the objects F(i, j) are zero whenever j - i > 2

A chain complex in a stable category gives you, in particular, a filtered object, and as Lurie discusses, the objects F(i, j) for general i and j are related to the higher pages of the resulting spectral sequence

Yeah, I'm sure something like that is possible

But if you pursue those coherences, you must end up writing down something like Gap

But if you're in the nerve of a 1-category, then you don't need the coherences

That much makes sense in any pointed oo-category, I guess

You can write down a sequence of objects in a stable oo-category with boundary maps such that d^2 = 0, but then there are a bunch of coherences you have to put in, because d^3 otherwise might be zero in two different ways

Yeah, I think that's exactly right

which is in D(Ab), not (Ab)

It seems from 1.2.2.3 that if your complex is ...->C_n -> C_{n - 1} -> ..., then if F is the corresponding object of Gap, F(n - 1 , n) = C_n[n]

If you start with a complex in an abelian category, though, I think your Gap object will take values in the derived category

That shows how an object of Gap gives you something recognizable as a chain complex

I think you want Remark 1.2.2.3

No, I could have figured that out

Ah, right

Of which work?

Where does he talk about Gap?

well, the arxiv seems to be back

Which is the same thing as product-preserving presheaves

It's the nonabelian derived category - freely generated under sifted colimits

Right

I think the connected Z/n-covering spaces correspond to the surjective morphisms \pi_1(X) -> Z/n

Hasn't the discourse up till now been that that isn't possible?

It looks like he gives a moduli-theoretic interpretation of Morava E-theory in terms of a theory of formal groups on commutative ring spectra

I don't know how plausible that seems to me, but some rich and powerful people seem to want to make it happen, because there was a link to the breakthrough prizes from the main google search page the other day

@JonathanBeardsley there is a potential utilitarian calculus here. It could be argued that if the breakthrough prizes really "arrive" in the public consciousness, they could raise the profile of math enough to draw vastly more funding than their cost

and someone, I think Tyler, recently told me there's one known as "the multi-author paper" too

there are two classical papers in homotopy theory that get referred to simply as "the five-author paper" and "the six-author paper"

Or actually, must a quotient of cohomological Mackey functors be cohomological?

That's true, the homology Mackey functors have to be cohomological

Do you know for sure that those two things are different?

Ah, I think what I was half-remembering is that a generic A_oo structure on Morava K-theory isn't homotopy commutative

Oh sure

I think Angeltveit also shows that a "generic" A_oo structure on Morava K-theory doesn't extend to an E_2 structure

there's always the stable module category, which is secretly a category of module spectra but you don't have to think about it that way

It looks like it's a livestream of a memorial event for Mirzakhani at Stanford

definitely not

Surely a representable functor from spectra to spectra is linear?

I have the strong impression there are very few, but I don't have a precise answer. I'd also like to know

Which is some kind of cyclotomic object, and then you could lift it to the corresponding TC

One might guess that the topological analog of the trace lands in the cohomology of the free loop space

aren't those two groups equal by definition?

presumably examples of non-left-complete t-structure come from stabilizing non-hypercomplete topoi or something

right

that's exactly the argument I was going to give that spectra are right complete, so I don't know why I didn't notice that it works both ways

good - sorry for misleading with my mistake earlier

ah, right

but maps between pro-objects aren't necessarily the same as maps between the limits, right?

if so, is that, in fact, true?

ok, let's see - is it the case that left completeness is equivalent to the mapping space between the Postnikov systems being equivalent to the mapping space of spectra?