Terminology question for everyone: it seems natural to me to call a sheaf of ∞-groupoids (in the healthy homotopy-coherent sense, of course) a 'stack'. I'd like to have a similarly unornamented word for a sheaf of (∞,1)-categories. (I've grown weary of prefixing every other word with an unsearchable string of symbols.) I saw that Charles Rezk used 'pile' in his ICM talk. Is this ... good? What are my options?
@DylanWilson Thanks for clarifications. A fiber sequence Free(1)->A->L_A[k] for E_k algebras A is given on Lurie's Higher Algebra (Theorem 7.3.5.1). It seems clear, but I did not realize it on my first reading of these materials. Sorry for such a stupid question.
I have no intuition on cyclotomic structures on THH(R). I have noticed this paper, but I even failed to understand how cyclotomic structure on THH(A) has something to do with Frobenius on A when A is an F_p-alebra intuitively from that aspect.
@ClarkBarwick I believe Kapranov (?) has used the word "schober" for this, which is apparently a (South or Austrian) German word for a "sheaf" in the agricultural sense...
What I mean is, aren't schobers supposed to satisfy descent only up to splitting localisation sequences of categories? I had in mind that schobers were something like sheaves valued in the fissile derived category ....
I'm not making any claims as to which word is best, I just wanted to point out the spelling in some of the existing literature on schobers in algebraic geometry :)
@DenisNardin I know that this has something to do with p times convering S^1->S^1. I know that in the paper of Nikolaus and Scholze, they claim that the cyclotomic maps are induced by Tate-valued Frobenii. I meant that I cannot put together the picture here and the paper I cited (geometry of cyclotomic trace), from which I can have a very vague understanding of the cyclotomic trace.
Hmm... I'm not super familiar with the perspective in that paper so I'll avoid making an ass of myself. My understanding of the cyclotomic trace is more "classical": I see it as a map of functors lifting the map $Σ^∞_+iC→\mathrm{colim}_{x\in iC} End(x)$ that sends each object $x$ to the identity in $Ω^∞End(x)$
Also, "Frobenii"? Frobenius is a German last name, not a Latin word :)
Well, I saw this word on Krause and Nikolaus' lecture notes, both the authors of which are German. It could be a typo, but I googled and found that the word also appears on stacksproject.
So phonetically it's closer to the Latin 4th declension? Also re: the sheaf of (oo, 1)-categories naming convention, I don't mind a small adornment if it allows for future generalizations. So I like, for instance, 2-stack.
I guess I was thinking of 2-stack in the sense of "the category of 2-stacks forms an (oo,2)-category". And I guess this was going with the convention that "oo-stack" means "(oo,1)-stack", as oo-category means (oo,1)-category. But I'd happily be convinced that this isn't the best terminology!
Let $X$ be a contractible topological space with a continuous action of a discrete group G s.t. all of the stabilizers are finite. Is $X/G$ (the actual quotient space not the homotopy quotient) rationally equivalent to $BG$?
It was my impression that there was no notion of "rationalization of $X$" unless $\pi_1 X$ was nilpotent, where you can filter it by things with abelian quotient and rationalize some Postnikov-type tower. But I don't really know this stuff.
That's the thing for which Bousfield invented Bousfield localizations
The reference is his The localization of spaces with respect to homology (not to be confused with the paper with the almost exact same title but about spectra)
Anyway, to prove X/G→BG is a homology equivalence, if I recall correctly there's a spectral sequence of the form $E^1_{p,q}=⨁_{\sigma\in X_q/G}H_p(St_G(\sigma);A)⇒H_{p+q}(X/G;A)$. Then the map X→EG induces an equivalence for the E^2-page when A=Q and all stabilizers are finite
there is a classical theorem of vietoris when everything in sight is compact (metric?) that a map with acyclic fibers is a homology isomorphism; it has a nice sheafy proof
The spectral sequence should come from the double complex $C_*(X;A)⊗_{\mathbb{Z}[G]} P_*$ where $P_*$ is a free resolution of $\mathbb{Z}$ as a $\mathbb{Z}[G]$-module, and we're taking simplicial chains, although the exact details elude me a bit here
Ben Antieau and David Gepner use "sheaf" to mean a sheaf of infinity-groupoids and "stack" to mean a sheaf of infinity-categories. I think it's compelling in the sense that infinity-groupoids play the role of sets, but it has the weird quirk that a lot of things we used to call stacks should now be called sheaves
@BrianShin That's really incompatible with the existing algebraic geometry literature (where they use "stack" to mean a sheaf of groupoids and "sheaf" to mean a sheaf of sets). There are plenty of theorems saying "the stack XXX is a sheaf"
I'm leaning towards "pile" or "schober" or some other made up word
I wanted to work it out carefully at some point to make sure I understood how it works, but well I never got around to do it
Ah I was remembering it wrong. The SS is $E^1_{p,q}=⨁_{\sigma\in X_q/G} H_p(St_G(\sigma); A)⇒H_{p+q}(X_{hG};A)$. Still it's good enough to run the argument you want
And of course what I wrote did not make sense. The homology of the total complex of $C_*(X;A)⊗_{\mathbb{Z}[G]}P_*$ is clearly $H_*(X_{hG})$
Its a cool fact. I just came to think about it from by trying to understand fundamentally why rational homotopy of $M_g$ is controlled by the mapping class group. This statement + teichmuller theory seems to make this intuition precise.
