i think so? somehow i always have gotten the sense that there's a canonical notion of "module over an O-algebra", even if different ones also exist

certainly there's a canonical notion for O=Comm, and for E_n i'd expect this corresponds to swiss cheese

@Aaron: so, for example, when O = Ass there are at least three reasonably canonical notions of module one could think about: left modules, right modules, and bimodules. there's a general definition on the nLab that I think spits out bimodules

@AaronMazel-Gee well, I still think this particular extension will exist, but nevermind: there's an easier way to get the example, do it in spaces! You get a social space whose image in simplicial objects in the homotopy category has no realization.

Autocorrect ought to correct, not make me say things like "social space"

@QiaochuYuan i have never opened it, but Fresse has a springer volume called Modules over operads and functors which is reputed to be about such things

@AaronMazel-Gee As for examples of that, does computing things like Tor or Ext for specific modules using specific resolutions count? I certainly wouldn't know how to do some calculations otherwise, but maybe that's just me.

Also, here's something simple that I think might be in the spirit of the examples you want. It's not something that uses any model category theory, but it is a hocolim I don't know how to calculate without building an explicit space that models it. Let I be a free category (i.e., free on some directed graph) and let G:I-->Groups be a functor that takes all morphisms of I to injective group homomorphisms. Then hocolim BG is 1-truncated. (This is theorem 1B.11 in Hatcher.)

@QiaochuYuan Thank you so much! Your answer on my chern weil question is spectacular! Can you recommend any specific sources that discuss these ideas?

In 2. I meant $Sym(\frak{g^*})^G=Sym(\frak{t^*})^W$.

nevermind my latex got scrambled.

I know that for derived / triangulated categories (, there is a "fill"-in map that is not in general unique - Can one say something that all fill ins induce the same map on homology, if we're talking about chain complexes? I am unsure, but I'm leaning towards no.

@Geoffroy: thanks, that's exactly what I was looking for!

@Saal: unfortunately I don't know any good refs. this is all stuff i picked up from a combination of MO and my advisor

there's this story about how the chern character is related to derived loop spaces that is in the literature in various places but i've never worked through it myself

one thing i can point you to is a more abstract description of chern-weil theory: in arxiv.org/abs/1301.5959 freed and hopkins carefully explain the sense in which chern-weil theory computes the de rham cohomology of the stack BG_{conn} of G-bundles with connection

(whose geometric realization is BG, since the space of connections is contractible)

@QiaochuYuan Thanks. What do you mean by geometric realization of a stack? Does this mean treating it as a simplicial object in some simplicial model category of presheaves and taking the realization to get a presheaf? (then hopefully you get something representable?...)

@Saal: I think a construction that works is: treat it as a simplicial stack on smooth manifolds (we need this to talk about connections), then there's a cosimplicial object in smooth manifolds you can restrict along to get a bisimplicial set, then take the geometric realization of the totalization of that

but i haven't really thought about it

Is it done in that paper you linked to?

Actually it's a question that bothered me anyway. For instance how go I get from the topological stack $[*/G]$ to the classifying space $BG$?

basically the same construction i think. for a topological stack you can even restrict to the usual cosimplicial object. this has the effect of taking the "singular simplicial set" of your stack, and in particular if you apply this construction to the presheaf represented by an actual space you get the geometric realization of its singular simplicial set

@QiaochuYuan Ah, great!

thanks

Just to clear the air, I didn't give a topics class this semester.

Does anyone know, given a diagram of E_n-ring (spectra) A<----R---->B, with pushout C, is it true that Mod(C) is the pushout (or pullback?) of Mod(A)<----Mod(R)--->Mod(B)?