I think that resolution in Prop 4.6 is using some finite generation hypothesis -- the standard cosimplicial object would be Hom_R(R[GxGx...xG],M), but since the domain is finitely generated free this can be identified with Hom_R(R[GxGx...xG],R) tensor M.
Say we want to work out C-comodule structures on X, for some coring C. If I take the cosimplicial object C-->CxC-->CxCxC-->... shouldn't Hom(X,C^\bullet x X) compute such structures?
It seems like the maps in the cosimplicial object are precisely the ones determining associativity and unitality.
Where C^\bullet x X is the aforementioned cosimplicial object tensored with X
if i take $C = R\otimes_S R$ and I'm working over $R$-modules, then when I smash with $X$ (over $R$!) I get $X\to X\otimes_S R\to X\otimes_S R\otimes_S R\to\cdots$
Hey @CraigWesterland! Not Toda brackets, but in Massey products this paper by Isaksen shows what can go wrong if your brackets only contain 0: arxiv.org/pdf/1309.4637v1.pdf
When working with ring spectra (with working here, I just mean using them as objects for generalized cohomology theories in somewhat rudimentary terms) how important is it to know details on how the smash product is constructed?
are there any general properties of ring spectra one should know about? For example, it bothers me somewhat that I haven't done any concrete calculations with ring spectra as one does for say rings, but maybe that isn't possible to do in the same way here?
Maybe it's important to know about what doesn't work for ring spectra. For instance, taking the cone on multiplication by 2 on the sphere spectrum does not give you a ring spectrum back.
Maybe a good thing to check out is the back of Ravenel's orange book where he shows how, for nice ring spectra, you can get so-called Hopf algebroids from the pair (R, R\wedge R) and so forth.
If you're going to learn chromatic homotopy theory, this is a good thing to be familiar with.
i recently learned the following fact: every spectrum can be written as the colimit of an ind-system of desuspensions of suspension spectra, and the smash product of two spectra so expressed is the colimit of the smash product of the two systems
i really appreciate how this points out where the cross-interference in, like, HF2 ^ HF2 comes from
it's also a sentence a model category neanderthal can still parse
also it's worth pointing out that this is the smash product defined in switzer, he's just reluctant to say the words 'ind-system' or 'homotopy colimit', and so it takes him a few pages to spit it out
it's sort of understandable; it's easy to be flippant about definitions in a chatroom, but when writing a textbook you're under some obligation to be thorough, and if you thoroughly express the above idea you end up writing switzer's book without much deviation
the disappointing thing about switzer is that it's more encyclopedic than it is conversational. you won't find these sorts of summarizing sentences anywhere in there
@Tedar: I am equally an amatuer but if you want some words on modern approaches to the smash product then Rezk's Notes on the Hopkins--Miller theorem and the start of Hopkins Notes on Elliptic Cohomology are both pleasingly short and informative
oh, don't. the point is that nobody knows how to abstractly characterize which maps I --> R of a spectrum into a ring spectrum ought to have that the cofiber R/I is again a ring spectrum
@AaronMazel-Gee yeah, though i think there have been mumblings that it's not really the right idea. they're called smith ideals, so, really i should say that it's jeff smith's ideals
it is (equivalent to) The Right Thing to define Spec of an E_\infty-ring spectrum to be a space with a sheaf of E_\infty-rings, namely the space Spec(\pi_0(R)), with the sheaf given by the various localizations R[f^{-1}]
@JonBeardsley i feel strongly that once everything is sufficiently categorical, then you get the right notions
I am quite curious if Lurie's DAG will have any applications to arithmetic geometry since I have mostly done that before - it would be interesting . But once again, I know nothing about DAG apart from some buzzwords
Well, consider cohomology theories for schemes of finite type over Spec Z and the conjectures related to L-functions / zeta-functions. There have been many attempt sand dreams to find "right" cohomology theories for proving these conjectures
so there is Weil-étale cohomology (see the work of Baptiste Morin, I remember that he said something he stumbled upon probably would be an (inf,1)-topos ) , and the dream of (the very, very) conjectural Deninger Cohomology
I think so too :) I know that some people , to realize these different theories are trying to generalize scheme theory in some sense for arithmetic schemes
yeah. so, i guess, right. as long as i've got a model category, i should be alright.
Finally, under extra conditions on R, A, and V , we describe fibrant replacements in MVA in terms of a generalized cobar construction. - Hess and Shipley
Hm. I think I'm getting somewhere.
u guys are the bee's knees
hopefully someday someone will come in here and have a question that I can answer... instead of the other way around.
i mean.... yeah. i dunno. i feel like i realized all this shit pretty on, but from a purely theoretical viewpoint. i had to give a talk on model categories, i dunno, my second year (i didn't know much topology when i got to graduate school, lol), and i realized like, oh shit, cofibrant replacement = projective resolution = "cell object" etc
but... what the hell to DO with that when your objects aren't modules....
@JonBeardsley I got the impression that that was because "take the (co)bar complex" often works. Which is I guess the point of that comment from Hess-Shipley you made earlier
@JonBeardsley left/right is incredibly important! this is just what i said previously. what matters is whether your functor of interest (e.g. cotensoring with some fixed comodule) is a left or a right adjoint
Maybe this is completely wrong. But if you actually want to compute these thing, there is probably going to be a spectral sequence Cotor(X_*,Y_*) -> Cotor(X,Y), in analogue with EKMM
