7:53 AM
@FrankScience I have been confused about several related things in the past -- however, Nikolaus-Scholze seem to indicate that the Tate diagonal is lax symmetric monoidal. where have you found that these don't commute?
regarding cosimplicial commutative things vs filtered commutative things, the two are indeed different
here's one example. if you have a commutative monoid in filtered objects, then the E_1-page of the associated spectral sequence is graded-commutative
e.g.: say A is a commutative Hopf algebra over k -- its cobar complex C(k,A,k) is then a cosimplicial commutative ring. in the E_1-term, the 1-line consists of equivalence classes [x] where x is primitive
the product of [x] and [y] is [x|y] in the cobar complex, which is not the same as [y|x]. the two become equal at E_2 because their difference is the boundary of [xy]
more generally, if you have a commutative filtered object ... -> A_2 -> A_1 -> A_0, the Day convolution gives you, e.g., (A_n ⊗ A_n) / Sigma_2 -> A_{2n}.
the same isn't true if you take a cosimplicial commutative object and convert it into a filtered object.
you get a map (A_n⊗A_n)/Sigma_2 -> A_n. it so happens that the map A_n⊗A_n -> A_n has a lift to A_{2n} -- but not the whole homotopy orbit.
in general the map from the homotopy orbit is kind of "spread out" between different layers of the filtration between n and 2n.
this isn't an "oh, things are better if you work coherently" thing -- it's fundamental to the Dold-Kan correspondence.
now, having said that, it's easy to get confused here because sometimes people conflate two different situations
first and foremost, it is an equivalence of categories between nonnegative cochain complexes and cosimplicial objects
second, it is also an equivalence of homotopy theories between (nonnegative cochain complexes + quasi-isos) and (cosimplicial objects + weak equivs)
in the first case, the dold-kan correspondence is lax symmetric monoidal in one direction (using the eilenberg-zilber shuffle map) and only lax monoidal in the other (using the alexander-whitney map), full stop
when you add in the homotopy theory, the eilenberg-zilber map becomes a natural weak equivalence. so you get a symmetric monoidal equivalence of homotopy theories between cochain complexes and cosimplicial abelian groups.
but the high-powered version of the Dold-Kan correspondence for stable categories isn't a generalization of the second case. it's a generalization of the first case
(meaning, e.g., you have an equivalence between cosimplicial spectra and filtered spectra, but on both sides the equivalences are defined objectwise, not some extra structure analogous to weak equivalences of complexes)
Alice Hedenlund has some notes on the filtered approach to spectral sequences here: math.ru.nl/~sagave/east2018/tate_construction_talk_EAST2018.pdf
(a) the lax-symmetric-monoidal eilenberg-zilber map is known to lift, so that every commutative object in filtered spectra becomes a commutative object in cosimplicial spectra.
(b) i am not aware of anybody who has managed to get the alexander-whitney map to lift to the coherent setting, which is what you would need to assert that the other half of D-K is lax monoidal. i thought i had an argument once, but it foundered on my lack of ability to make the technical details go (eg I don't know how to make "Stasheff associahedra" arguments for imposing associative algebra structures in the new framework)
(c) random, related: why didn't anybody ever tell me that the alexander-whitney map is geometric? if you take the subspace of (Δ^n x Δ^n) consisting of the union of (front p-face) x (back q-face), that's a subspace homeomorphic to Δ^n
4 hours later…
12:13 PM
1:02 PM
@TylerLawson Thanks very much for clarifying these things! I had a "program" to think about this problem, but never concentrated on it too much. My first sanity check would have been to show that cosimplicial -> filtered is lax monoidal - but I did not try this either.
I wonder if one can make sense of an operad E^fil (in the classiacl sense) valued in filtered spaces, where E^fil(n) is the space E Sigma_n, with its usual skeletal filtration. Then one should take algebras in filtered spectra for this operad, call the category FAlg. (1) Does CAlg(cosimplicial) map naturally to FAlg? (2) Can FAlg be described more intrinsically?
@DenisNardin right, so: let's call Delta_{/[n]} = K (the "stasheff associahedron") and its boudary B; you get a map from the pushout of K <- B -> Delta^op_{\leq n-1} to Delta^op_{\leq n}.
1:25 PM
@TylerLawson I have noticed that on BMS18, they exhibited a cyclotomic structure on $\DeclareMathOperator\THH{THH}\THH(A/\mathbb S[z])$ when A is a (commutative, I don't know whether this is necessary) $\mathbb S[z]$-algebra, where $\mathbb S[z]$ is the free $\mathbb E_\infty$-algebra with a single generator $z$.
@TylerLawson I think it works better by taking K=(Δ_{\le n})_{/[n]}? I don't think it is straightforward, but I think it is the same sort of argument that shows it is a categorical equivalence (as in Bousfield-Kan). I've tried to write it down for a couple of minutes and I failed, though, so it might harder than I thought (Lurie does talk about the associahedron in HA, but he uses a completely different method)
Yeah, I see why: the multiplication map is just induced by the map $n→*$, so it can be factored $n→S^1→*$ where the first map is just embedding n points into $S^1$
1:41 PM
@FrankScience this is a point that I have been confused on myself in the past (to the point of making public errors when I state things).
- the Tate construction, and the Tate diagonal, are also invariant under the C_p-action on X⊗X⊗...⊗X: true
e.g., the multiplication (X⊗X)^{tC_2} ⊗ (Y⊗Y)^{tC_2} -> (X⊗Y⊗X⊗Y)^{tC_2} requires knowledge of which factor of X gets multiplied by which factor of Y.
2 hours later…
3:45 PM
@TylerLawson It seems that what you said also applies to the space level version - which I thought was correct. Given a commutative monoid M, we consider two compositions $\DeclareMathOperator\Bcyc{B^{\mathrm cyc}}\Bcyc M\to M\to M^{hC_p}$ and $\Bcyc M\to(\Bcyc M)^{hC_p}\to M^{hC_p}$ which don't coincide for general commutative monoids $M$?
3 hours later…
6:29 PM
@FrankScience If it's a genuinely commuting monoid, I believe that you are OK because you can factor through fixed elements, rather than homotopy fixed elements. But I think that you are correct in the same objection does apply to E_infty spaces.
e.g.: let's say C is a symmetric monoidal category. if we have a map B^{cyc}(C) -> D of categories, we in particular have a functor F: C -> D together with a natural isomorphism T: F(A⊗B) -> F(B⊗A). it factors through the augmentation B^{cyc}(C) -> C if and only if T is F(twist).
now, the Tate diagonal B^{cyc}(C) -> B^{cyc}(C)^{hC_2} -> C^{hC_2} has the structure of such a functor.
if we trace it explicitly, we have C x C -> (C^4)^{hC_2}, then the two functors (C^4)^{hC_2} -> (C^2)^{hC_2}, and then the multiplication (C^2)^{hC_2} -> C^{hC_2}.
this composite sends (A,B) first to (A,B,A,B) with the permutation isomorphism; then the two functors send it to (A⊗B,A⊗B) with the flip iso and (B⊗A, B⊗A) with the flip iso; then it maps to A⊗B⊗A⊗B, and the natural isomorphism T is the two-fold twist.
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