Who made that Adams spectral sequence calculator? It's really nice!

@AlexanderCampbell I missed the seminar but watching it now

very nice! They succeeded!!

Did we get any word on when this paper will be available?

@HarryGindi Cisinski said in the talk that they are writing a paper. I forget if he indicated how soon it will be available.

I'll email Kim since I know him haha

btw are you doing any online seminar things?

like, apart from this one

Also, I was surprised you didn't talk about your paper at MSRI, since you had earlier results, but we can talk about that in private

I'm speaking in the MSRI seminar this week msri.org/seminars/24958

the ∞,2-coherent nerve

cool

I spoke about my nerve paper in a seminar on (∞,2)-categories at MSRI back in February.

if you can remember to remind me before you go live by email, I definitely want to come (I'll also set a calendar thingy)

(But I'm glad that you at least remember the order of events.)

Will do

Well, I was watching it with some friends, and I said something like 'oh, why isn't Alexander speaking about it?'

=D

@PaulVanKoughnett it's a joint project between dexter and hood chatham, & it is super duper nice

@HarryGindi Actually, as my talk will be at 7am Sydney time, I probably won't remember to remind you.

Ah, where will the zoom link be?

There's no link on the seminar page

@DenisNardin Is there a stack that I'm secretly working with here? It seems if I was working with a stack (in groupoids) $\mathfrak{X}$ and an element $x\in\mathfrak{X}$, then we have that for any $u_1,u_2\in \mathfrak{X}(U)$, that $(g:V\to U)\mapsto \hom_{\mathfrak{X}(V)}(g^*u_1,g^*u_2)$ is a sheaf, so in particular, $(g:V\to U)\mapsto \hom_{\mathfrak{X}(V)}(g^*x,g^*x)=\text{Aut}_{\mathfrak{X}(V)}(g^*x)$ is a sheaf (since it's a groupoid). But in my case, I don't think you get a stack in (cont)

groupoids by taking, say, $\mathfrak{X}(U\to X)$ to be the category of $\mathcal{O}_U$-modules for each $U\to X$, and pullbacks for morphisms.

I guess to make it clearer what I'm asking. 1. You said 'in general this works for any stack', which made me think that twisted forms for sheaves of $\mathcal{O}_X$-modules (for some subcanonical topology) can be treated stack-theoretically, can they? 2. What does it mean to twist something by something else? I.e. if $\mathcal{F}'$ is a sheaf of $\mathcal{O}_X$-modules is a twisted form of $\mathcal{F}$, do we phrase this as $\mathcal{F}'$ being twisted by something?

@GaloisintheField Why not? Do you have a counterexample? In any case quasi-coherent $\mathcal{O}_X$-modules form a stack

@DenisNardin Oh, well, it was just the groupoid part that I was confused about. But perhaps you just define the collections of morphisms, to only have the isomorphisms?

To answer the more general case: if $x\in\mathscr{X}(U)$ is an element of a stack, a twisted form for $x$ is another element $y\in\mathscr{X}(U)$ that is locally isomorphic to $x$, that is to say that there exists a cover for your topology $\{\varphi_i:U_i\to U\}$ and isomorphisms $\varphi_i^*x\cong \varphi_i^*y$

@GaloisintheField that's correct, you have to ignore all non-iso's to get a stack

@GaloisintheField Yes, you take only the isomorphisms. Although it makes also sense to talk about stacks of categories, but that's not what I was referring to

Okay, I think I'm happy with that then. I'm still not sure about the semantics of 'twisting' an element of a stack by torsors for some group-sheaf

@DenisNardin To continue, the general theorem tells you that there is an equivalence between the groupoid of twisted forms of $x$ and the groupoid of torsors over $\mathrm{Aut}_{\mathscr{X}}(x)$ that sends a twisted form $y$ to the sheaf $\mathrm{Iso}_{\mathscr{X}}(x,y)$ with the right action given by precomposition

The inverse is given by sending a torsor $T$ to the object that we can denote by $T\times_{\mathrm{Aut}_{\mathscr{X}}(x)}x$, although defining it precisely takes a bit of work

Essentially you pick a cocycle representing $T$ and you use it to "twist" the gluing data of $x$ to the gluing data of a twisted form

Informally we say that we are twisting $x$ by $T$ to get a twisted form $y$

Excellent, thanks! That makes things much clearer

The only thing you need to be careful about is that "schemes over S" do not form a stack: that's why sometimes the twisted form of a scheme is going to be an algebraic space (although this never happens for quasi-projective schemes)

Indeed an algebraic space is precisely a sheaf that is étale-locally represented by a scheme

@DexterChua Thank you for the response! I don't know how to work with Toda brackets, but it's clear how Z/4 follows from the Ext chart so I'm happy.

and, wow, this computer program looks amazing! I tried to play with it a little, but I don't understand the input file format very well, even after poking around some of the source code on Github. Is there documentation anywhere I can take a look at?

(I was able to edit a downloaded file to get the program to display the E2 page for the Joker over A(1), but further modifications fail when I try to add classes in degrees >= 5. No doubt this is me misunderstanding how to specify the input data)