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:**

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

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 ~~ this is entirely correct. *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.

**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...

over$C$; the duality just switches the role of the section and the retract. I've just noted this in the question. So e.g. over $\ast \in \mathrm{Top}$, one thinks about cogroup objects in $\mathrm{Top}_\ast$ -- e.g. the pinch map on $S^n$ is an example-up-to-homotopy. But it does appear that the grouplike condition is very strong for comonoids in $\mathrm{Set}_{X/-/X}$ -- I think $\mathrm{Ab}(\mathrm{Set}_{X/-/X})$ is always trivial. $\endgroup$8more comments