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Apr
20
comment Is this a model for $K$-theory of a triangulated category?
I'd be surprised if this was a Kan complex since (as I mentioned in the other post) it's not known whether every triangulated category is n-angulated for all n. (For example, this would imply that every pre-triangulated category is automatically triangulated, i.e. that the octahedral axiom is redundant. That would surprise most people, I think.)
Apr
19
comment Difficulties with descent data as homotopy limit of image of Čech nerve
There is no such in HTT. I remember trying to find a reference for this and failing a while ago, but have dim memories of Lurie getting around this problem a different way... I'll see if I can find it.
Apr
18
comment Difficulties with descent data as homotopy limit of image of Čech nerve
For (1) I think the essential point is that the inclusion of $\Delta_{\le n}$ into $\Delta$ is "n-(co?)final" in the sense that the $\Delta_{\le n} \times_{\Delta} \Delta_{/[m]}$ has an $n$-connected nerve for all $m$. So the analog of Quillen's theorem A says that computing limits over one or the other should be the same.
Apr
18
comment Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
The diagram is produced algorithmically. If you want to do it for n composable morphisms, draw a them in a line. Now extend the line to the right with a zero, and down on the left with a zero. Now fill in this rectangle with pushout diagrams (so you're looking at an [n+1] \times [1] diagram.) Now on the far lower right you can extend by a map to 0, and on the lower left offset by one you can extend by zero... rinse wash and repeat. You get a staircase lookin' thing.
Apr
18
comment Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
(It is not known whether any triangulated thing is n-angulated, but any stable $\infty$-category is $n$-angulated for any $n$, and the proof is the same as in loc. cit.)
Apr
18
comment Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
the picture at the bottom of p.24 here: math.harvard.edu/~lurie/papers/HA.pdf has an evident generalization to any n-tuple of morphisms. If you write it down you'll join the group of m people who have independently discovered the notion of an "n-angulated category". The shape you seek comes from collapsing that unrolled version in some probably not-so-trivial way.
Apr
18
comment Lower central series quotients in terms of (co)homology
Fix a prime. The p-lower series has associated graded a restricted Lie algebra and we get a spectral sequence starting from restricted Lie algebra cohomology and converging to group cohomology. Restricted Lie algebra cohomology can in turn be computed from a spectral sequence starting with ordinary Lie algebra cohomology (which can be computed with a Koszul complex). I wonder of your observations (for a p-group) follow from low-degree considerations of this set-up, while higher degrees will be messier in general because there could be room for differentials?
Apr
15
comment Two transfers for ramified or branched covers
(For example, oriented theories usually don't have Gysin maps for ramified covers. Any cohomology theory that does splits as copies of eilenberg MacLane spectra).
Apr
15
comment Two transfers for ramified or branched covers
My hesitation here is that the pushforward I just defined for cohomology is not obviously related to the Gysin map for proper, oriented maps (which seems to be what you're claiming). Do you have a reference for that fact or am I misunderstanding?
Apr
15
comment Are all formal schemes *really* Ind-schemes?
I think without finiteness hypotheses you're doomed? There's a careful discussion of this stuff in the stacks project, where they discuss what conditions you need to have this nice Ind-presentation you're looking for. stacks.math.columbia.edu/tag/0AIT seems relevant, but I guess they don't focus much on morphisms
Apr
15
asked Two transfers for ramified or branched covers
Apr
2
comment Chern classes of PU(n)-bundles
Ah- apologies, I guess I spoke too quickly. So then I guess something like first $n^2-n$ will be nonzero and the rest will be zero?
Apr
1
comment Is $MGL$ an $H\mathbb{Z}$-algebra?
Certainly not... For the same reason MU is not an HZ algebra.
Mar
30
comment Chern classes of PU(n)-bundles
Indeed, every symmetric polynomial except the last one has at least some monomial that doesn't contain one of these diagonal elements! (Which is essentially the content of the last paragraph).
Mar
30
comment Chern classes of PU(n)-bundles
It's not the case that each monomial has such an element. For example, the first symmetric polynomial is just the sum of all the $x_{i,j}$. Most of those will map to something nonzero.
Mar
27
comment Sheaf on a filtered topological space?
It's still not clear what it is you're asking. Are you asking "what is a constructible sheaf"? Or "can I revover a constructible sheaf from its value on strata?" (No.) Or "what information do I need in addition to its values on strata?" (You need the gluing data. For example, if there's only one proper closed stratum, then you need (1) a locally constant sheaf on the closed guy, (2) a locally constant sheaf on the open guy, and (3) a map from the sheaf on the closed guy to the `germ of the open part on the closed part', i.e. what you get by pushing forward then restricting to the closed bit.)
Mar
25
comment What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
The way Simpson actually states his conjecture is to say that he thinks that's what the correct parts in Kapranov-Voevodsky actually prove. So Simpson's conjecture is a guess of an answer to your question of what can be salvaged.
Mar
25
comment Chern classes of PU(n)-bundles
What do you mean? Ad* takes x_ij to t_i-t_j and Chern classes go to the corresponding symmetric polynomials.
Mar
25
comment Useful, non-trivial general theorems about morphisms of schemes
How about something like Chevalley's theorem? It says that a quasi-compact morphism which is locally of finite type preserves locally constructible sets.
Mar
24
comment Stability of adjunctions of infinity-categories by base change
Ah! Fair enough