Homotopy Theory

11:58 AM

@DenisNardin I do believe that it is only a matter of sign if M is connected, because the dual map is a map S[M]-->S[pt], which seem to have very few options if M is connected (we want to hit the projection to the point map), and it splits out as a summand with complement having no maps to S[pt] when M is connected. So if we are happy with the fundamental class up to sign its perfectly fine.

but its probably too late for your lecture right? :-)

Bryan Shih

3:31 PM

Hi all,

Context: I've been thinking about the tower argument Lemma 2.21 https://arxiv.org/pdf/1803.10897.pdf in this paper (again, where I have previously asked) and figured I'm unclear about the following fact:

What is a criteria for checking if an object is prozero in Pro(Fun(C,Sp)), pro category of Fun(C,Sp)? It seems to me that the functora \pi_i :Pro(Fun(C,Sp)) -> Pro(Fun(C,Ab)) forms jointly conservative map?

If so, how does one see this?

Edit: prerhaps a more fundamental question: I'd like to understand hwo one checks an object is prozero in Pro(Sp).

Tom Bachmann

Pro(C) is compactly cogenerated by C, so I suspect a pro-object (X_i) is zero iff for every T in C, colim_i Map(X_i, T) = 0

Iirc pro homotopy groups do not give a conservative family

someone more up to speed with the intricacies might answer better in a bit :)

Denis Nardin

@TomBachmann Indeed: the classical example is the difference between a constant pro-system at a spectrum and the Postnikov tower of that spectrum, seen as a pro-system

They have the same homotopy groups, but they corepresent radically different objects ($\operatorname{Map}(E,-)$ and $\operatorname*{colim}_n \operatorname{Map}(t_{\le n}E,-)$ respectively)

@BryanShih I don't think anyone understands that, beyond what Tom says :)

Bryan Shih

3:58 PM

Thanks Tom and Denis. Now I am extremely confused. How exactly does one make sense of the last line of lemma 2.21 's argument?

A. "If each postnikov section of {F_r}_r a proobject in Fun(C,Sp) is pro-zero"
B. "{F_r} itself is, by devissage"

I don't know how to check A.

And for B. let us write F_i as lim_n t_{\le n} F_i, I get stuck in pulling the llimit out:
colim_i Map (lim_n t_{\le n} F_i, T)
if we write F_i as colim_n t_{\ge n} F_i this argument doesn't work either: we cannot commute with limits and colimits.

Denis Nardin

4:29 PM

@BryanShih Oh that's because the objects they're working with are pro-truncated (that is every object in the pro-system is n-truncated for some n). Among such objects you can check equivalences on homotopy groups

Indeed the map I described earlier is the universal pro-truncation map, and you can see that if E is itself truncated it is an equivalence

Bryan Shih

4:47 PM

@DenisNardin , sorry, i still dont see why you chan check equivalences on homotopy groups from your argument

I agree however that if E is truncated, that it is equivalent to {t_{\le n} E }

Denis Nardin

@BryanShih Ok, let me give you a full argument. Do you agree that pro-spaces are product-preserving functors Space→Space?

Bryan Shih

Yes

Denis Nardin

Then I claim that a pro-space is protruncated iff it is right Kan extended from Space_t, the subcategory of truncated spaces

Or, said it differently that a map of pro-truncated pro-spaces X→Y is an equivalence iff for every truncated space Map(X,T)←Map(Y,T) is an equivalence

Therefore this is the same as asking that the map t_{\le n}X→t_{\le n}Y is an equivalence for every n

But then you can go by induction using the fact that it's an equivalence on all Postnikov layers

(this is actually easier for spectra than spaces, since I don't have to worry about basepoints)

(here I'm using that if T is an n-truncated space, and X is a space Map(X,T)=Map(t_{\le n}X,T) )

Bryan Shih

So does the statement of second line follow from first?

Denis Nardin

pro-truncated pro-spaces are cogenerated by truncated spaces, by definition

Bryan Shih

5:02 PM

Ok: but why can we check the map t_{\le n} X -> t_{\le n}Y is an equivaence by homotopy groups?

Denis Nardin

Well you do it by induction on n. For n it's true by definition π_0=t_{\le0}

Bryan Shih

My thoughts of applying induction is by consider fiber/cofiber but this doesn't seem to lower the homotopy groups?

Denis Nardin

5:18 PM

Well, the point is that a map of Eilenberg-MacLane spectra/space is an equivalence iff it is an equivalence on homotopy groups, pretty much by definition

Or rather, what does it mean to be an equivalence on homotopy groups?

Bryan Shih

To me: it forms a jointly conservative map to ptd sets

Denis Nardin

I am confused

What's your definition of homotopy groups of a pro-spectrum?

Bryan Shih

Oh, I misinterpreted your question

Denis Nardin

(let's use spectra because basepoint issues become tricky)

Bryan Shih

I thought yuo meant homotopy group for spaces

but homotopy groups for pro Spectra is the induced map from Spectra -Ab. So for each j it sends {X_i} -> \pi_j {X_i }

Denis Nardin

5:24 PM

Ok, do you agree that the functor pro-Ab→pro-Spectra sending {A_i} to {HA_i} is fully faithful?

Because now I'm going to claim that if E is a pro-spectrum we have a natural fiber sequence Σ^n Hπ_nE→t_{\le n}E→t_{\le n-1}E for every n

Bryan Shih

yes, i agree it is ff , and I also agree with the fiber sequence

Denis Nardin

Ok, therefore a map of n-truncated spectra is an equivalence iff it is an equivalence on π_n and on the (n-1)-truncation, do you agree?

Or, I guess, since we are using bounded above stuff, using cotruncations may make the proof easier

You don't need this since your pro-spectra are also bounded below, but I actually also claim that if E is a pro-spectrum we have E=colim_n t_{\ge n}E

As before you can check this by hitting it with Map(-,T) for T a spectrum and using that colimits commute with colimits

Bryan Shih

I feel something lost in your first statement:
Is the gneeral claim here that if we have a diagram of fiber sequence
A_0->B_0->C_0
A_1->B_1->C_1
with vertical maps going down, the outer two being equivallences, then the middle is equivallence?

Denis Nardin

Well, in a stable category, yeah

(that's part of why I wanted to leave pro-spaces...)

Bryan Shih

I agree in stable category, and great now we can use induction!

Wow, this has been really helpful

thanks a lot!

Denis Nardin

5:36 PM

No problem! Pro-objects are super tricky, I am often lost working on them...

Aaron Mazel-Gee

7:17 PM

this is belated, but thank you everyone for the comments on covers and sieves -- it's been very helpful!

Jonathan Beardsley

10:45 PM

I think I've gotten myself confused by op's again. If I've got a symmetric monoidal ∞-category C^⊗→Fin∗, then I can naively take opposites to get a Cartesian fibration (C^⊗)^{op}→(Fin∗)^{op}. Am I right in understanding that this is the Cartesian fibration classifying the (unique) symmetric monoidal structure on C^{op} induced by that of C?

Piotr Pstrągowski

Suppose that I have a functor $A: R-Alg \rightarrow Ab$ from $R$-algebras into abelian groups (a "discrete stack") which is isomorphic to a formal group after a faithfully flat extension (that it, it is then isomorphic to the formal spectrum of a power series ring). Then is A then a formal group itself?

Jonathan Beardsley

And then if I want to get a coCartesian fibration describing the symmetric monoidal structure on C^{op} I can take the dual of the opposite fibration described above, (C^{op})™→Fin∗.

Because on the other hand it seems like I can do the following: take the dual (C^⊗)™→Fin∗ which is a Cartesian fibration classifying the monoidal structure on C, and then take its op ((C^⊗)™)^{op}→Fin∗. This gives me a coCartesian fibration which, again, I think describes the symmetric monoidal structure C^{op}.

So basically, does "dual" commute with "op"?

Ah sorry... see I've definitely messed up an "op" in there somewhere....

Piotr Pstrągowski

I made mistake: in what I wrote above I should have written "is Zariski locally isomorphic to the formal spectrum of a power series ring". Thus, I'm asking if having this property fpqc-locally means that it also holds in the Zariski topology.

Jonathan Beardsley

11:04 PM

But anyway... yeah if I start with a cocartesian fibration describing monoidal structure on C, then I can take the opposite Cartesian fibration of it, and then take the dual, and this should describe the monoidal structure on C^{op}. On the other hand I can take the dual first, to get a Cartesian fibration and then take the opposite coCartesian fibration, and I think this should ALSO describe the monoidal structure on C^{op}.

Ah.... I guess this Theorem 1.7 of Denis, Clark and Saul's paper on dual fibrations.

Namely, that commutative diagram.

So given symmetric monoidal ∞-categories p:C^⊗→Fin∗, q:D^⊗→Fin∗, can I say that a functor F:C→D is oplax monoidal exactly when it corresponds to a functor (C^⊗)^{op}→(D^⊗)^{op} that preserves (opposites of) Cartesian lifts of inert morphisms in Fin∗?

@RuneHaugseng I'm actually trying to puzzle out your comment in your paper:

I agree that an oplax monoidal functor should be a functor (C_⊗)^op→(D_⊗)^op that preserves (cocartesian lifts of) inert morphisms.

But it seems like a functor C_⊗→D_⊗ preserving Cartesian lifts of inert morphisms would be lax monoidal, since those Cartesian fibrations classify the monoidal structure on C, not on C^{op}?

This is the commutative diagram from Saul, Denis and Clark's paper:

Jonathan Beardsley

11:35 PM

No... somehow I've messed this up again lol.

Sorry I've totally pushed @PiotrPstrągowski's good question into oblivion with my nonsense question. Please scroll up to see it.

Rune Haugseng

11:55 PM

@JonathanBeardsley Not sure I understand your question - as you say you can dualize and take op in either order and you get the same thing. And a functor (C_⊗)^op→(D_⊗)^op that preserves cocartesian morphisms over inerts gives precisely a functor C_⊗→D_⊗ preserving Cartesian lifts of inert morphisms when you apply op (to the whole fibration, not fibrewise)

Jonathan Beardsley

@RuneHaugseng yeah I just messed up the ops. Figured it out now. Sorry to ping you.

Rune Haugseng

This is oplax monoidal since for x, y in C you have a cartesian morphism x \otimes y -> (x,y) in C_\otimes that maps to F(x \otimes y) -> (Fx,Fy) in D_\otimes (where the target has that form since F preserves inert cartesian morphisms). That's not necessarily cartesian in D_\otimes, but its cartesian factorization gives a morphism F(x \otimes y) -> Fx \otimes Fy in D.

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