Homotopy Theory

Saal Hardali

00:32

What is "homology with local coefficients" really?

Cohomology with local coefficients is just sheaf cohomology which is just global sections (of sheaves of abelian groups or spectra), but what is homology?

In some cases homology is a special case of cohomology. For example: for a suitable class of spaces (Which include manifolds and varieties but am unsure what's the most general statement - hopefully finite type CW complexes) there's verdier duality which gives homology as a special class of cohomology.

correction - "special case" - first line in the third paragraph.

sorry third line in second paragraph

Denis Nardin

@SaalHardali As you say, for reasonable spaces you can see E-homology as $f_!f^!E$ where $f:X\to *$ is the morphism to the one-point space. I am not sure if there is a good definition of homology for unreasonable spaces, but if there is, it has to be equivalent to the procedure I gave

What I meant is that for CW-complexes $f_!f^!E = E\wedge \Sigma^∞_+X$

I meant if $X$ is a manifold $f_!f^!E=E\wedge \Sigma^∞_+X$, sorry

Saal Hardali

00:50

Wait, is $E$ a sheaf of spectra? Where is this smash product taking place then?

ah nevermind I got it.

It doesn't answer the question though

yours is a definition for homology with constant coefficients

Denis Nardin

To get homology with non-constant coefficients you just take a sheaf of $f^!E$-modules that is locally free of rank 1

Saal Hardali

And then push it with $f_!$?

Denis Nardin

Yes. I think that if X is not a manifold you have to mutter something related to the dualizing complex (a.k.a $f^!S$)

Saal Hardali

I'm really tempted to believe at this point that there's a coommon generalization of verdier duality and Spanier–Whitehead duality somewhere here

Denis Nardin

You are using SW duality in this (overly general) form of Verdier duality

Saal Hardali

00:57

Aha so it is

And $f^!S$ is the SW dual?

Denis Nardin

What do you mean?

Saal Hardali

I don't know exactly I'm asking. I've never seen such a general form of Verdier duality

Perhaps this is why I had no idea what is homology with local coefficients in the first place

Actually It would help me even if the case of HZ module is somehow explained.

Denis Nardin

So, that case is quite explicit

For example f^!HZ is the sheaf of Borel-Moore chains

Then it is clear that f_!f^!HZ is the complex of compactly supported B-M chains, a.k.a. C_*(X;Z)

Saal Hardali

Hmmm nice. What I'm trying to understand is this:
Suppose $F$ is a sheaf of $HZ$ modules. What does it mean to take homology with coefficients in $F$. I can calculate it via a procedure I learned way back but what is the meaning of these groups?

Denis Nardin

I am not sure I know how to define "homology with coefficients in HZ"

Saal Hardali

01:03

This is my point!

Wait what.

Homology in HZ just means homotopy groups of $HZ \wedge \sigma^{\infty}_+ X$?

Denis Nardin

Uh I mean in coefficients in F, sorry

Saal Hardali

Ah ok so now were on the same page

You said there's a verdier duality though so you might know how to define them (albeit in a not very direct way)?

Denis Nardin

One thing that happens is that (let's make everything HZ-linear) D(f_*F)=f_!(DF)

Saal Hardali

Ok this has to happen for it to be called verdier duality.

Denis Nardin

Where D=F(-,f^!HZ) is the Verdier dual

So in particular f_!F is dual to f_*(DF) and so I might be tempted to call it the homology of F

Going by general principles that the homology should be the cohomology of the dual, at least for dualizable objects

Saal Hardali

01:09

That's what I meant.

Denis Nardin

However f_!F is usually called the compactly supported cohomology so I am slightly uncomfortable calling it the homology

Saal Hardali

But I'm not so sure that the procedure I know calculates this thing even for very basic examples.

Denis Nardin

What's the procedure you know?

The only definition I know for local coefficients is when you have a local system of Z-modules

Saal Hardali

Yeah. It's messy

I don't like it

Denis Nardin

It's not, you can form the corresponding sheaf of B-M chains and that is a f^!HZ-module, which is the kind of thing I am comfortable taking homology of

Moreover I strongly suspect that this is just the Verdier dual of the sheaf computing the cohomology of the dual local system

I mean, the concrete definition with the explicit chains is fairly intuitive, isn't it?

Saal Hardali

01:16

Say $F$ is the sheaf of $HZ$ modules we can say $f_*\mathcal{Hom}(F,f^!HZ)=\mathcal{Hom}(f_!F,HZ)$

Is this what you meant?

Denis Nardin

The point is that Verdier duality works in situations where you can work interchangeably with homology and compactly supported cohomology

Saal Hardali

Yeah, this is what confuses me about homology with local coefficients. I understand there's a procedure and it's not very far fetched (nothing surprising). But somehow I can't find a way to justify it conceptually

In particular I can't "compose" functors that compute homology like you can compose sheaf cohomology (push $\circ$ push)

As a result there's a lot less to work with.

That's why I like having things conceptual

Denis Nardin

So, usually local coefficients are useful when you work homotopically. In that case a local system is just a functor $F:X\to HZ-Mod$ and homology is just the colimit and cohomology is just the limit

Saal Hardali

Ah so that's better already. Why haven't I thought about that.

In general a sheaf of spectra is just a functor $X \to Sp$ isn't it?

Denis Nardin

A local system of spectra, not a general sheaf

Saal Hardali

01:24

Yeah sorry abuse of terminology

I mean a homotopic sheaf

Or am I missing some crucial point here...

Isn't everything locally constant for infinity groupoids?

Denis Nardin

Well, I would never talk of a sheaf over an ∞-groupoid. It is bad terminology (although it has a meaning, it is simply confusing)

Saal Hardali

Sorry. Will not do this in the future.

So it is true that everything is locally constant over $\infty$-groupoids right?

Denis Nardin

So, a true statement is that a locally constant sheaf over a space is the same thing as a functor out of the corresponding ∞-groupoid (although it is a nontrivial theorem).

I am not sure which definition of "sheaf over an ∞-groupoid" you are using

Saal Hardali

Not sure what's the most concise way to say this but a "sheaf" of spaces for instance in my earlier terminology would just mean an $\infty$-groupoid over the my $\infty$-groupoid.

I'm told that one can stabilize the over category of an infinity groupoid to get parametrized spectra.

Denis Nardin

So, it turns out that this is equivalent to a functor from your ∞-groupoid to the category of ∞-groupoids. It is very much a special thing though. I think this terminology is more confusing than anything else and you should ditch it, but do as you feel comfortable

Saal Hardali

01:33

Already ditched it :)

Denis Nardin

Parametrized spectra again are local systems of spectra, not sheaves of spectra

Saal Hardali

Good Good.

So sheaves of spectra are something you say when there's a site right?

Denis Nardin

Yes. In general speaking of sheaves where there's not site (or topos) in sight is confusing

Saal Hardali

Got it. Thanks!

Although there is the obvious site for an infinity groupoid which gives the correct answer right?

The over category I mean.

Denis Nardin

Yes (that is a topos, not a site). That's why it is not strictly speaking an abuse of terminology

Saal Hardali

01:35

Sorry mean topos

Denis Nardin

I wouldn't recommend using it though

Saal Hardali

Yeah yeah, just wanted to make sure we were on the same page

I need to go to sleep now (4 am lol). Thanks for the help! If you have some epiphany about this homology thing please tag me here thanks!

Charles Rezk

15:16

Here's a question: some people (including me!) talk about (n,r)-categories, meaning n-categories in which all k-morphisms with k>r are invertible. Where does this terminology come from? Who invented it?

Bruno Stonek

16:39

Let $R\to T$ be a morphism of commutative S-algebras. If I have a T-algebra A, what is the relationship between THH^T(A) and THH^R(A) ? I'm particularly interested in the case where $R\to T$ is some Bousfield localization of $R$ at some spectrum $E$ (smashing, if this helps). This sounds like a classical problem but I can't seem to find much about it (even in the algebraic HH case). Perhaps I'm lacking some keyword

Geoffroy Horel

@BrunoStonek If R\to T is aTHH etale extension they are the same. The THH etale condition means that THH^R(T) is equivalent to T

In general they are related by equation 8 of arxiv.org/pdf/math/0306243v1.pdf

taking X=S^1 and using your notation you get the equation $THH^T(A)=THH^R(A)otimes_{THH^R(T)}T$

Denis Nardin

Smashing localizations are THH-étale, right?

Geoffroy Horel

16:54

so if T is obtained from R by a Bousfield loc L that is smashing

then THH^R(T) is L-locally equivalent to T

but since they are both local they are equivaelnt

does this sound correct ?

Denis Nardin

I think so. I was thinking more crudely, something like the simplicial diagram in the definition of THH^R(T) is (homotopically) constant

Geoffroy Horel

yes I agree

Bruno Stonek

this seems to be the argument given by McCarthy-Minasian between remark 3.4 and example 3.5

@GeoffroyHorel thanks! it seems like the keyword I was lacking was "(thh)-étale"

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