@JohnGreenwood Assuming these are 2-periodic, in general the E_0-homology of E (or K) will be the ring parameterizing isomorphisms from the formal group of E to that of E (or K), by Landweber exactness.
If K and E are associated to the Honda formal group law over F_{p^n}, then the E_0 homology of K will be the ring of continuous functions on the Morava stabilizer group valued in K_0; the K_0 homology of K will be the same, but with some extra exterior elements; and the completed E_0 homology of E will be the same, but with continuous functions valued in E_0.
One place to read about this is Hovey - Operations and cooperations in Morava E-theory.
IIRC, if K and E are associated to a formal group G over a field k<F = (algebraic closure of Fp), and S = group of pairs (f : k ~ k, a : G ~ f G defined over F), then Gal_k(F) acts on S, and the K_0 homology of E is continuous Gal_k(F)-equivariant functions on S with values in F.
Given a co-cartesian fibration $E \to X$ classifying a functor $F: X \to Cat_{\infty}$ I can pass to the left fibration $E^{cart} \to X$ consisting of the cocartesian edges in the total space. Does this operation model the same thing as composing the functor $F$ with the internal groupoid functor to get a functor $(F)^{\cong}: X \to \mathcal{S}$?
@SaalHardali Yes. To prove it just unstraighten the inclusion $F^\cong→F$ to a map $E_0→E$ of cocartesian fibrations. Since the lhs is a left fibration, it factors $E_0→E^{cocart}$ and it induces an equivalence on every fiber
Going a while back in a chat we discussed this simplicial set $Q$ classifying cartesian fibrations (which Nygen proves in his thesis is a quasicategory). Assuming this is the quasi-category of infinity categories we have the natural inclusion of the right fibrations (which cisinski already proved is the correct model) and apprently this construction gives the adjoint on the nose
You get a map of simplicial sets $Q \to S$ which models the adjoint of the inclusion on the nose.
I wonder how far can these ideas be pushed to simplify the basic theory.
@DenisNardin Thanks for the reply. Sorry for my ignorance on the subject, but whats HR for a Green functor? By the way, is there any standard reference on Green functors? There's nothing on the nLab page.
@user40276 I just meant the Eilenberg-MacLane spectrum (which is a lax symmetric monoidal functor from Mackey functors to G-spectra). Dunno about a standard reference, but this should cover some material
For a simple example, the derived category of Mackey functors can be identified with the category of HA-modules in G-spectra, where A is the Burnside Mackey functor sending G/H to A(G/H)
@user40276 Yes, precisely. If you interpret G-spectra as spectral Mackey functors, this is just postcomposition with the classical functor from abelian groups to spectra
@DenisNardin Ah, ok. You were considering ordinary Mackey functors. So you are just picking functors from the effective Burnside category on both sides. I'm unsure, though, on whether the category complexes of Z-modules (for Z as a Mackey functor) is equivalent to the derived category of Mackey functors in the sense of say Kaledin, but whatever. Thanks for the help.
Here's a question: take the category of abelian group objects in $G$-spaces (or in $G$-simplicial sets if you prefer), and give it a model structure so that $A\to A'$ is a weak equivalence if $A^H\to A'^H$ is for each $H\leq G$. What is this?
I suspect it is $\infty$-equivalent to the connective derived category of cohomological Mackey functors. Is something like this known?
@user40276 No, Kaledin's derived category is equivalent to $HZ⊗S$-modules (i.e. the "linearization" of the sphere G-spectrum), which is a completely different ring
@SaalHardali Which theory do you want to simplify? I'd say it (conceptually simplifies) straightening/unstraightening.
To each simplicial set $S$ we have a set (functors $\mathfrak{C}(S) \to \underline{QCat}$), where $\underline{QCat}$ is the Kan-enriched category of quasicategories. Lurie shows that this is representable by a quasicategory $Cat_\infty=N(\underline{QCat})$.
For each simplicial set $S$ we also have a set (cocartesian fibrations over $S$). Nuyen shows that this is (more-or-less) represented by a quasicategory $Q$.
Lurie's unstraightening construction, applied to $\mathfrak{C}(N(\underline{QCat}))\to \underline{QCat}$, produces an explicit cocartesian fibration over $Cat_\infty$, and hence gives rise to an explicit map $u\colon Cat_\infty\to Q$.
This map $u$ represents the unstraightening construction.
Presumably $u$ is an equivalence, so it has a (non-explicit, but unique up to contractible choice) inverse $s\colon Q\to Cat_\infty$, which "represents" straightening.
I doubt that the underlying work to prove all this is much simpler than what Lurie does. But it seems conceptually cleaner to me.
I suppose what I wonder is whether the Cisinski-Nguyen theory can be used to entirely circumvent Lurie's straightening / unstraightening -- do all your infinity-category theory without ever touching simplicial categories.
One thing about $Cat_\infty$ is that, as a quasicategory, it contains the 1-category of quasicategories. So st/unst is ultimately the thing that turns 1-categorical diagrams in the category of quasicategories into (co)cartesian fibrations.
So if you care about using the 1-category of quasicategories in this way, you need something.
Let $C$ be the 1-category of quasicategories. I wonder how hard it is to write down an explicit map $N(C) \to Q$ which models the functor $C \to Cat_\infty$
It should be possible, since $Q$ is Joyal-fibrant.
By $C \to Cat_\infty$ I mean the canonical inclusion of simplicial categories, where $C$ is locally discrete.
We have one, namely $N(C)\to N(\underline{QCat})=Cat_\infty\to Q$.
The funny thing about $u$ I should have mentioned is that, although $Cat_\infty$ and $Q$ have the "same" objects (quasicategories), $u$ is not the identity map on objects.
Perhaps you want an $N(C)\to Q$ which is on objects the (restriction of the) identity map. I seem to remember Lurie does some special version of straightening/unstraightening for 1-categories which does this (perhaps).
I'm trying to think if there's anywhere in the theory of presentable infinity-categories maybe where straightening / unstraightening has to be explicitly invoked. Maybe to solve some coherence problem.
Anything involving limits or colimits in $Cat_\infty$ uses st/unst (as in 3.3.3-4).
Often tho, Lurie works hard to remove explicit dependence on st/unst when he can. For instance 5.2.1. Unfortunately, this means using the model category of marked simplicial sets, which has its own annoyances.
@CharlesRezk Perhaps "simplifying" isn't the correct word. I was thinking about disentangling the basics of the theory of $\infty$-categories (by which i'm mainly referring to chapters 3-5 in HTT) from simplicial categories.
Just the idea that the $\infty$-category theory of quasi-categories can be developed in a self contained way without the need to refer to other models appeals to me. Just like it's nice that the homotopy theory of Kan complexes can be developed completely independently from topological spaces though they are in the end equivalent.
@CharlesRezk Also regarding what you said about S/U before. Isn't there also an explicit map going in the other direction $Q \to Cat_{\infty}$ which is given on $n$-simplices by straightning cartesian fibrations $\phi : X \to \Delta^n$ to functors $St_{\Delta^n}(\phi) : \mathfrak{C}[\Delta^n] \to Cat_{\infty}$?
I didn't have enough time to edit. In the end there I meant the simplicial category $\underline{Set^{+}_{\Delta}}$ and not $Cat_{\infty}$.
(also for the correct variance I should have said cocartesian fibrations there...)
Straightening $\phi\colon X\to \Delta^n$ gives you an enriched functor $\mathfrak(\Delta^n)\to \underline{sSet}$, to the simplicially-enriched category of (marked?) simplicial sets. But I don't think the output is necessarily quasicategory-valued, so you have to apply a further fibrant replacement, which is kinda inexplicit. Do I have that right?
@CharlesRezk Yeah, I agree, that's what I missed. Thank you for pointing it out (for the second time I feel like...). This does complicate things. If we were doing the same with left/right fibrations and functors to spaces we could use $Ex^{\infty}$ to get an explicit functor in the other direction. Here there may be nothing we can do to get an explicit functor.... i did miss this point.
@CharlesRezk Doesn't the left adjoint to the relative nerve functor give a more explicit "straightening" of a quasi-category over Delta^n to a functor Delta^n --> qCat?
Scratch that. One still of course needs to take a pointwise fibrant replacement.
