What are the tangent $\infty$-categories to $\mathrm{Top}^\mathrm{op}$ and $E_\infty$-$\mathrm{Ring}^\mathrm{op}$?

Question

Given an $\infty$-category $\mathcal{C}$ with finite limits and finite colimits, there are two ways to make it stable. One is to pass to the pointed objects and take the category of $\Omega$ objects $\mathrm{Sp}(\mathcal{C}_\ast)$. An $\Omega$ object is a sequence of pointed objects with equivalences $X_n = \Omega X_{n+1}$. Or dually, we pass to co-pointed objects and take $\mathrm{Sp}^\vee(\mathcal C^\ast) = \mathrm{Sp}((\mathcal{C}^\mathrm{op})_\ast)^\mathrm{op}$ the category of $\Sigma$-objects. In terms of the original cateogry $\mathcal{C}$, a $\Sigma$-object is a sequence with equivalences $X_n = \Sigma X_{n+1}$. (We could also mix and match (co)pointing and loops/suspension -- I'd be interested to hear about this too.)

This works in families, too: the tangent $\infty$-category $T\mathcal{C}$ of $\mathcal{C}$ is the category fibered over $\mathcal{C}$ whose fiber $T_C\mathcal{C}$ at $C \in \mathcal{C}$ is the stabilization $\mathrm{Sp}((\mathcal{C}_{/C})_\ast)$ of the slice. We could also take the dual tangent category $T^\vee \mathcal{C} = T(\mathcal{C}^\mathrm{op})^\mathrm{op}$ (I don't want to call it the cotangent category, since there's already something different called the cotangent complex). Note that the result of the pointing / co-pointing procedure is the same in both cases, yielding the category $\mathcal{C}_{C/-/C}$ of split objects over $C \in \mathcal{C}$. That is, $T_C \mathcal{C} = \mathrm{Sp}(\mathcal{C}_{C/-/C})$, while $T^\vee_C\mathcal{C} = \mathrm{Sp}^\vee(\mathcal{C}_{C/-/C})$. The difference likes in the spectrum procedure.

Now, the two foremost interesting examples of this construction are when $\mathcal{C} = \mathrm{Top}$, where $T\mathcal{C}$ is the category of parameterized spectra, and $\mathcal{C} = E_\infty$-$\mathrm{Ring}$ the category of $E_\infty$ ring spectra, where $T\mathcal{C}$ is the category whose objects are modules over $E_\infty$ rings, and morphisms are composites of ring homomorphisms and module homomorphisms (i.e. the Grothendieck construction applied to the functor $\mathrm{Mod}: E_\infty$-$\mathrm{Ring}^\mathrm{op} \to \mathrm{Cat}$ sending a ring to its category of modules).

The funny thing is, I'm more inclined to think of $\mathrm{Top}$ as being analogous to $E_\infty$-$\mathrm{Ring}^\mathrm{op}$ ("derived affine schemes") than to $E_\infty$-$\mathrm{Ring}$ for "geometric" purposes. So it seems strange to apply the same, non-self-dual construction to both categories. So I ask:

Questions:

1. If $X$ is a space, what is the category $\mathrm{Sp}^\vee(\mathrm{Top}_{X/-/X})$? Is it somehow a category of modules?

2. If $R$ is an $E_\infty$ ring, what is the category $\mathrm{Sp}^\vee(R$-$\mathrm{Alg}_\mathrm{aug})$? Is it somehow a category of derived schemes parameterized over $\mathrm{Spec R}$?

For (1), I should note that when $X$ is the empty space $\emptyset$, so that we're actually asking about the (opposite of the) stabilization of $\mathrm{Top}^\mathrm{op}$ as in my question title, the answer is that the stabilization is the terminal category, because $\emptyset$ is a strict initial object in $\mathrm{Top}$, and the first step of stabilization is to take pointed objects. But this also happens when one computes the stabilization of $E_\infty$-$\mathrm{Ring}$, so apparently this is no obstacle to the construction being interesting at other slices. The question is what a ``$\Sigma$-spectrum over $X$" looks like. When $X$ is a point, this is again trivial because a space which is an $n$-fold suspension for every $n$ is contractible. I think this will also happen whenever $X$ is simply-connected? But still, this may be interesting over other $X$'s.

For (2), one likewise needs to understand suspension in $R$-$\mathrm{Alg}_\mathrm{aug}$, where $R$ is an $E_\infty$ ring. The temptation is to say that this is given by topological Hochschild homology, but not so fast -- in augmented $R$-algebras, THH is the tensoring over $\mathrm{Top}_\ast$ with $S^1_+$, whereas suspension is tensoring with $S^1$ itself. So I don't know what the suspension of an augmented $E_\infty$-algebra is. this is entirely correct.

EDIT So a $\Sigma$-object in $R$-$\mathrm{Alg}_\mathrm{aug}$ is an augmented $R$-algebra $A_0$ equipped with the data of infinite "de-THH'ing": for each $n \in \mathbb{N}$, there is an $A_n$ with $THH_R(A_{n+1}) \simeq A_n$. Is anything known about this kind of thing? It would be amazing to have some analog of infinite loopspace theory to recognize when this can be done... It makes me think of iterated algebraic K-theory, and also about redshift...

Answer 1

If $X$ is any space, then here is a consequence of the Blakers-Massey excision theorem:

If $(X \to Y \to X) \in Top_{X/-/X}$ is such that the map $Y \to X$ is $n$-connected, then in the suspension $X \to \Sigma_X Y \to X$ the map $\Sigma_X Y \to X$ is $(n+1)$-connected.

(Technically, the proof for $n=0$ is a special case not covered by Blakers-Massey.)

As a corollary of this, if we have an object in $Sp^\vee(Top_{X / - / X})$ consisting of augmented objects $Y_k$ with equivalences $Y_k \simeq \Sigma_X Y_{k+1}$, the map $Y_k \to X$ are automatically $n$-connected for all $n$ and hence $Y_k \to X$ is a weak equivalence.

Said another way, all objects in $Sp^\vee(Top_{X / - / X})$ are equivalent to the zero object.

The ring case is a lot more interesting (but subject to a little bit of unpleasantness because homotopy limits don't usually commute across $\Sigma_R$). When $R=H\Bbb Q$, for example, we can embed the rational homotopy theory of pointed 1-connected spaces into the category of augmented $H\Bbb Q$-algebras, and this carries a (1-connected, finite type) rational spectrum $X$ to an object in $Sp^\vee(H\Bbb Q-Alg_{aug})$, which can recover its cochain object $C^*(X;\Bbb Q)$.

Answer 2

As indicated in the original question, an object in $Sp^{\vee}(\mathcal C_*)$ would be a $\Sigma$-object: a sequence of objects with weak equivalences $X_k \simeq \Sigma X_{k+1}$, and this is not so interesting when $\mathcal C = Top$.

Ah ... but what if one changes the notion of weak equivalence? In particular, localize $Top$ like Bousfield likes to do: fix a prime $p$, and let $L_n^f$ be localization with respect to a map $\Sigma A \rightarrow *$, where $A$ is a finite complex whose suspension spectrum has type $n+1$ and whose mod $p$ homology has connectivity as small as possible. Then I think that the category of $\Sigma$ objects is roughly the same thing as the category of $K(n)$--local spectra (maybe subject to correction due to the telescope conjecture), with the correspondence being: given such a spectrum $X$, there is an $L_n^f\Sigma$-object with $k$th space $\Theta_n(\Sigma^{-k}X)$, where $\Theta_n$ is the `left adjoint' to the better known telescopic functor $\Phi_n: Top \rightarrow Spectra$.

[See my 2008 HHA paper for these constructions and references.]

I think that Brayton Gray made some use of this idea, when $n=1$, but I can't see anything about this on his publications listed on MathSciNet.