I had a bad experience where I had to wait about an hour for check-in because the check-in clerks obviously had absolutely no clue what they were doing
still, thousands of people must fly Spirit every day, and presumably nothing too dire happens to most of them
in reference to a distinction i made above, let me just make sure this sounds reasonable. so if you have a presheaf of sets considered as a presheaf of spaces, then to be a sheaf of sets is to be a sheaf of spaces, right? i feel like this should be because the sheaf condition is a (ho)lim condition, and this commutes with n-truncation
I have seen you guys talking about a certain dropbox - can one get access to it? (Well, obviously one can, but I would be interested in seeing what is in there)
@Tedar a quasiprivate collection of papers and notes, some harder to find than others. here's a current contents listing: f.cl.ly/items/1J2s0i412f0Q1O1G3V0X/… , send me an email at [email protected] if you'd like read access
Is there a categorical way to represent a comonoid, in the same way that we think of a monoid as a category with a single element and a collection of maps?
Yeah. Like, a set with a $G$ coaction is a functor $BG\to Set^{op}$
Hrm.
Ah, but okay, that kind of makes things make sense.... since a Hopf-algebra is really a group scheme. So, basically I do the stuff in the opposite category then apply global sections.
There was this half-dead ladybug on my desk, and I didn't know what to do with it. So I just threw it behind my desk. I can't sit here and watch it die. I'm weak.
Okay, yeah. So we have to do things internally, I guess. I.e. for a comonoid $M\in C$, i.e. a monoid $M^{op}\in C^{op}$, the comodules over it in $C$ are going to be modules over $M^{op}$ in $C^{op}$ yes?
Lord. I don't even remember why I started thinking about this.
Could anyone help me understand why the resolution appearing in proposition of 4.6 of math.harvard.edu/~lurie/papers/DAG-XI.pdf is the one for computing group cohomology?
I mean, I only know 2 ways to do this, either compute $Ext_{Z[G]}(Z,M)$ or like, this cosimplicial guy coming from $Hom(G^n,M)$
I don't remember what the second one was anymore, but it was something about comparing your cosimplicial thing with what Lurie writes right before that proposition
Well, that's what Lurie references for this particular section, hah.
Meh, they just give the standard definition. Sheds no light on why he calls that the resolution which computes group cohomology. Maybe it is just what Saul says, but I just don't understand why we're needlessly dualizing things.... erg.
Maybe that's not true actually, I just have to figure out what $\mathscr{A}(G)$ is.
But... what? I mean, we can always form the group ring over whatever our base is, and take coefficients in a $G$-module. So what's the generalization exactly?
Ah, $\mathscr{A}(G)$ is the pushforward of the structure sheaf of $G$ down to the base scheme.
And the cohomology of a group scheme $G$ over $S$, with coefficients in a sheaf $F$ can be defined as homology of the complex whose n'th level is given by $\Gamma(S, F\otimes \mathscr{A}(G)\otimes\cdots\otimes \mathscr{A}(G))$
Yeah.... I'm not sure, but I think this more general SGA framework is precisely what Lurie (and really Tyler as well) are using, but homotopically. It's probably exactly what Saul is saying, I'm just taking a long time to understand it.
@Jon: I'm late to the party, but a Hopf algebra is not a group scheme! commutative Hopf algebras are (the rings of functions on) affine group schemes, but noncommutative Hopf algebras are not even "noncommutative group schemes" in the sense that they are not group objects in Ring^op!
