Homotopy Theory

« first day (868 days earlier) ← previous day next day → last day (2535 days later) »

12:27 AM

(At some point I'll probably want this for QCoh and Mod in the derived sense but I'd be happy if there was an underived version too.)

Let X be a scheme over a base ring k. Let A be a k-algebra. The categories QCoh(X) and Mod(A) should be modules over the monoidal category Mod(k). I'm guessing there should be a notion of a tensor product Mod(A) ⊗_Mod(k) QCoh(X), which should be equivalent (as a module over Mod(A)) to QCoh(X_A) for X_A the base-change to A. Is there actually such a tensor product of module categories and is it easy to see the guessed equivalence?

Omar Antolín-Camarena

1:08 AM

Hi @JonBeardsley, I saw your question about converting a map $S^k \to GL_1(R)$ into a map $\Sigma^k R \to R$, but didn't get a chance to respond until now. This is confusing because of basepoints: $GL_1(R)$ is the union of some of the components of $\Omega^\infty$, those corresponding to units in $\pi_0 R$. The natural basepoint of $\Omega^\infty R$ is $0 \in R$ which corresponds to $0 \in Hom_R(R,R)$, but this is not present in $GL_1(R)$.

So if you have a map $f : S^k \to GL_1(R)$ landing in the component $u \in \pi_0(\Omega^\infty R))$, you can look at the translation $\tau_{-u} : \Omega…

I think Tobi and I never have arbitrary maps $S^k \to GL_1(R)$, but rather pointed ones (the natural basepoint of $GL_1(R)$ being 1), so $u=1$ for us.

This class of $\tau_{-1} \circ f$ is the thing we called the characteristic $\chi(f)$. Though now that I'm rereading the definition we wrote I have to say it's not clear at all.

I guess what happened when we wrote that definition of $\chi$ is this: The operation $f \mapsto \tau_{-1} \circ f$ induces maps $\pi_k(GL_1 R) \to \pi_k(\Omega^\infty R)$ and it probably felt like $f-1$ was weird notation for that map in the case $k>0$, so we just called it $f$ in that case (but kept $f-1$ for when $k=0$...

We should rewrite the definition to say what I said above.

Jon Beardsley

@OmarAntolín-Camarena Thanks so much Omar!

Hrm.... is there some obvious way to think about a map $S^n\to C$ for a quasicategory $C$?

Like... some reinterpretation of that data...

Qiaochu Yuan

1:26 AM

@Arpon: there are theorems to this effect in the derived sense in some papers by ben-zvi and nadler

the easiest version to think about is the underived affine version. here we consider, say, the 2-category of cocomplete presentable k-linear categories and cocontinuous k-linear functors. there is a universal property you could ask for of a tensor product, namely that it's universal for k-linear functors cocontinuous in each variable, and wrt this notion of tensor product we have Mod(A) otimes Mod(B) = Mod(A otimes_k B) by the universal property of the enriched yoneda embedding

pro

1:55 AM

@Arpon there exist both derived and underived versions of these and are basically an incarnation of Barr-Beck. For the derived version there is the paper of Ben-Zvi-Francis-Nadler (although I'm sure this particular contruction was already known to others, like Toen-Vezzosi, Orlov or some other russians down the street).

@Arpon For the underived version again there should be multiple sources. The one I know of is by Gaitsgory (notion of a category over an algebraic stack), who attributes to Deligne (again I'm sure there are some other belgians around who knew this already).

clarification: what is described in the paper of Gaitsgory is not a tensor product of abelian categories but rather a "base change" of QCoh along the ring map k --> A. In the derived world there exists a tensor product of stable categories, but in the abelian world I don't think there is one (or maybe there isn't a consensus on one, I forget, @Pieter knows more about this stuff).

Omar Antolín-Camarena

2:28 AM

I guess above I answered what I guessed your question really was ("what is this $\chi(f)$ supposed to be?") instead of what you literally asked, @JonBeardsley. As for the literal question I think basepoints get in the way again: your second method is OK, is by the isomorphism $\pi_k(GL_1 R) \to \pi_k(R) = \pi_k(\Omega^\infty R)$ you mean the one induced by $\tau_{-u}$; the first method you mention I don't think is well defined:

If $S^k \to GL_1R$ is not a based map, the best you can do with smashing with $R$ is get $\Sigma^\infty_+ S^k \wedge R \to \Sigma^\infty_+ GL_1R \wedge R$, but you really wanted the domain to not have the plus!

If $S^k \to GL_1R$ is based, then you could get a map $\Sigma^\infty R \to \Sigma^\infty GL_1(R) \wedge R$, but the action map can't be a map of spectra $\Sigma^\infty GL_1(R) \wedge R \to R$, because it's supposed to send $1 \wedge r \maspto r$, but 1 is the basepoint, so it should act by 0.

Does that make sense?

Jon Beardsley

2:50 AM

@OmarAntolín-Camarena absolutely! thanks so much for writing all of that!

I think it is true that I was trying to basically understand what $\chi(f)$ is.

How did you guys guess that these Thom spectra might also be versal algebras of some characteristic?

Sir Cumference - Pies

Howdy

3 hours later…

Omar Antolín-Camarena

6:05 AM

@JonBeardsley Chronologically the first thing we had in the paper was the proof of Mahowald's theorem about HF_2; that goes through identifying a Thom spectrum with a versal algebra, and makes the general case pretty clear,

Omar Antolín-Camarena

7:02 AM

In that special case the Thom spectrum is roughly S/(1=-1), while the versal algebra is roughly S/(2=0) --which are easy to accept as equivalent.

What I don't think would've been so easy to come up with is the idea of looking at the versal algebras at all; but we didn't do that, I heard about them from Jacob: when he told me about Mahowald's theorem he stated it as "HF_2 is S/2 in the world of E_2 algebras"; then I looked Mahowald's theorem up and saw people usually state it in terms of Thom spectra.

When I asked Jacob why the two formulations were equivalent he said it should be easy to prove that the versal algebra and the Thom spectrum represent the same functor in E_2-algebras.

Pieter

7:55 AM

@pro and @Arpon You have indeed several flavours of base change, on the abelian, triangulated, dg and even scarier levels.

When it comes to tensor products of abelian categories things get a little hairier apparently.

Deligne has indeed a tensor product, but it only works in special cases.

My office mate is working out a more general tensor product for Grothendieck abelian categories, that work should almost be finished now as far as I can tell. I don't know how base change fits into this, but I do know that tensor products of Qcoh of projective varieties corresponds to Qcoh of the fiber product, in her framework.

I'll ask her whether there is such a thing as an interpretation for base change.

Pieter

8:25 AM

She hasn't worked out the details of that application yet.

7 hours later…

Arpon

3:02 PM

@QiaochuYuan Thanks! I hadn't realized before you just said so the relationship between Mod(A) and enriched yoneda. Cool.

@pro @Pieter Thanks for the pointers.

2 hours later…

Justaskin_

4:44 PM

Anyone care to explain this to me: In the HoTT book, p199 (W-types), it states that List(A) has one nullary label (the empty list), and for each $a \in A$ a unary label corresponding to appending $a$ to the head of the list. But on p192 (the non-W definition of List(A)) it was defined using one nullary constructor and, for each a in A, a binary constructor cons: A \to List(A) \to List(A). Is p199 a misprint or am I misunderstanding the definition of a W-type?