When typesetting names for categories, I usually write $\mathsf {Cat} $, $\mathsf {sSet} $ etc. I think it looks good, but I haven't seen many others who use this font.
Question: What is the relationship between the plus construction on B(\Sigma_n \wr U(1)) (the wreath product of the symmetric group and U(1)) and the space BU(n)?
@user101036 Even if there is, surely he doesn't do anything that depends heavily on that choice, so you might as well take it to be the good one: Namely, the one presented by the twisted arrow category
So the idea would be to construct some kind of family version of the twisted arrow ∞-category with the property that for ever pair of objects x and y, the fiber is itself an excisive functor $\mathbf{Top}^{\mathrm{fin}}_\ast \to \mathbf{Top}$.
If we see spectra as excisive functors $S^{fin}_*\to S$ and we are in a stable category then I think that the mapping spectrum from $X$ to $Y$ is the functor $S\mapsto Map(X,S\otimes Y)$
Then I think you can describe this as the pullback of the twisted arrow category along the map $C^{op}\times C \times S^{fin}_*\to C^{op}\times C$ given by the tensoring
@AaronRoyer Fair point. It seems a bit as though one wants to pass to the twisted arrow category of the "Kleisli category" for the action of $\mathbf{Top}^{\mathrm{fin}}_\ast$ on C.
Oops, and there's Denis with essentially the same thought again. I'm not needed here.
(Oh, and in case it's not clear: that was called humor. I'm not really impossibly arrogant. (I recently learned that some mathematicians fail to understand irony or exaggeration unless it's underlined as such.))
@AaronRoyer Hmmm ... Let's see. I guess what you're after is a fibrational model using $C$ alone. I can sort of picture something like that for certain presheaf categories and then try to restrict. That's a bit hackneyed.
@JonBeardsley Maybe I should bring back the irony mark.
Things I've learned from Clark Barwick in the past 24 hours: what a Opilione is, what the irony mark is, and that $BGL_1(\mathbb{S})\simeq \mathbb{S}$-line.
@ClarkBarwick That does seem like more than you should need. I don't necessarily need/want a mapping spectrum in the sense of Higher Algebra spectrum objects. In fact, I feel like it might be easier to extract a fibrant symmetric spectrum, perhaps even using some good model for the Kleisli category, as you say.
This would just be the excisive functor you mentioned restricted to the spheres, but since you may not need an explicit choice of model for the $\infty$-category of spaces to make it work it feels to me more in line with the mapping spaces case
@AaronRoyer It's a good idea. My best guess for this (and as you can see, I haven't thought about this at all) would be to write down a model for a stable ∞-category with a "chosen" suspension. (Of course, up to a contractible choice this is no choice at all, but we're after models here ...)
I don't know if this is what you're getting at, but if you take a map of the form $(p,x):(K^{\rhd})^{\mathrm{op}}\times\Delta^0\to C^{\mathrm{op}}\times C$ in which $p$ is a colimit cone, and you pull the twisted arrow ∞-category back along this map, you get a left fibration over $K^{\lhd}$, and it satisfies condition (2) of Cor. 3.3.3.3 of HTT.
Why are there so few chances to edit? I should've called the map $(p^{\mathrm{op}},x)$.
I have a totally different question that has been bothering me for a while.
Say I have an $E_1$-ring $A$ over an $E_\infty$-ring $k$. Then the topological Hochschild cohomology of $A$ relative to $k$ is $k$-linearly dual to the topological Hochschild homology of $A$ relative to $k$.
The former is an $E_2$-ring in a natural way, which seems to me to imply that the latter should be naturally co-$E_2$. However, I only see a co-$E_1$-structure coming from the fact that it's a bar construction. Is there some way to see the co-$E_2$-structure?
I feel like it must come from the self-duality of $A \otimes_k A^{op}$, but I haven't had any luck making that even close to rigorous.
I had it in my mind that if you view $R$ as an $R^e$-$k$-bimodule, then it is right adjoint to $R^{op}$ as a $k$-$R^e$-bimodule. Then it follows from the description of THC as $Hom_{R^e}(R, R)$.
@JonBeardsley But if you're talking about $G$-$E_\infty$-algebras, we don't even know how ideal theory works for ordinary rings. Mike Hill knows what's known.
There is definitely something going on, since computationally you can see that THH is co-E_2 in more cases than you'd expect, I just now think that this is not the way to see what's happening.
@ClarkBarwick I think these sorts of things are perhaps more high-tech than he needs. I'm not sure. I think really what he wants is a notion of derived completion in an equivariant setting.
But I'm not sure. He works with things like Fredholme operators and stuff that I get very confused about very quickly.
I first noticed this when I was working with $THH^{\mathbb{Z}}(\mathbb{F}_p) \simeq \mathcal{O}(\mathcal{L}_{\mathbb{Z}} Spec \mathbb{F}_p) \simeq \mathcal{O}(\Omega^2_{(p)} Spec \mathbb{Z})$.
Do Kac-Moody groups have rigid representation theory? If not, equivariant K-theory isn't homotopy-invariant and I don't see how you expect to have spectra.
@ClarkBarwick Okay, but is equivariant topological K-theory one of these? I feel like you still need that representations don't deform to make your excision work.
Yeah, I'm not 100% clear about what he's trying to do. He wrote something on the board that looks like $[X,\mathcal{F}_0(\mathcal{H})]_{\mathbb{K}_J(A)}=I\mathbb{K}(X)$
@Sven I'm not Aaron, but in general ideals of ring spectra are not a good idea (there are a couple of competing definitions and none of them behave as you'd expect). What do you need them for? Maybe you just need ideals of the ring of coefficients?
whoa. I just proved that something had an adjoint, in real life, by checking the solution set condition. I never thought this would happen.
(nothing in sight was close to presentable, or the opposite of something presentable, nor obviously well-powered, etc... so I really needed to check it. And it worked! Weird...)
@DenisNardin I have a simple spectrum, the 0 space is fredholm operators on a sutable hilbert space with an action of a Kac-Moody group the rest is just loops on that. I have ideals of the fredholm operators (the 0 space) that I want to extend to the whole spectrum, in a nice way.
Ok, but then what do you want to do with that? Assume you have a map of spectra $I\to E$ that can be identified with "inclusion of an ideal", what do you need from it?
As I said there's not a good notion of "ideal of a ring spectrum" that works in every situation, so the particular notion you need to use depends on what you want from it.
It's for dominant K-theory, there is no notion of a K(A)-equivariant spectrum. All my maps are between spaces
Yea, I've gathered that over the last few hours of reading. I just started thinking about this today, I'm not exactly sure what I want my ideals to be.
K(A) being the K-M group generated by the matrix A, sorry for the confusing notation
It's localy compact, but not compact. It looks in many ways like a Lie group, but it's infinte dimensional, and all it's non-trivial represenations are infinite dimensional
