If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?

Question

I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.

We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos which admits a cover by the étale spectra of $E_{\infty}$-rings. The definition of a nonconnective spectral scheme, on the other hand, is a spectrally-ringed space which admits a cover by the Zariski spectra of $E_{\infty}$-rings. Suppose we instead look at spectrally-ringed topoi which are locally isomorphic to the Zariski topos of an $E_{\infty}$-ring. Clearly, these generalize nonconnective spectral schemes; and, on the other hand, these objects admit a natural functor to nonconnective spectral DM-stacks, which should be a fully faithful embedding extending the inclusion of schemes as schematic stacks.

My question is this: are these objects actually more general than nonconnective spectral schemes, or is the underlying topos always generated by a topological space? If the former, how much more general are they? (For instance, a nonconnective spectral DM-stack is schematic iff it admits a cover by $(-1)$-truncated affine objects. Is there a similar characterization of these "generalized schematic stacks"?)

Answer 1

Yes, they are more general. This is in fact already the case with ordinary rings. Let's call a classically-ringed $\infty$-topos which is locally the Zariski $\infty$-topos of an affine scheme an $\infty$-scheme. A classical scheme is then the same as a $0$-localic $\infty$-scheme. To construct an $\infty$-scheme which is not classical, let $F$ be any object in the Zariski $\infty$-topos $\mathrm{Shv}(X)$ of a classical scheme $X$. Then the slice $\infty$-topos $\mathrm{Shv}(X)_{/F}$ with the restricted sheaf of rings is an $\infty$-scheme (it is covered by open subschemes of $X$), which is classical iff $F$ is $0$-truncated. This same construction works to define spectral $\infty$-schemes that are not $0$-localic.

A theorem of Lurie (Theorem 2.3.13 in DAG V) implies that every $\infty$-scheme is such a slice over a $1$-localic $\infty$-scheme. I do not know if there exist $1$-localic examples that are not slices over classical schemes. If one works with the étale topology, however, this theorem implies that every DM $\infty$-stack is a slice over a classical DM stack.

$\infty$-schemes can also be identified with the full subcategory of $\mathrm{Shv}_{\mathrm{Zar}}(\mathrm{CRing}^\mathrm{op})$ consisting of those Zariski sheaves (of spaces) $F$ that admit an effective epimorphism from a coproduct of representable sheaves $R_i$ such that each $R_i\to F$ is "representable by slice $\infty$-topoi" (see Proposition 2.4.17(6) in loc. cit.). Unlike with classical schemes, however, such sheaves do not usually satisfy descent with respect to the étale topology, because higher Zariski and étale cohomology generally disagree.

ETA: This last remark implies that the natural functor from $\infty$-schemes to DM $\infty$-stacks is not fully faithful. If it were, then the functor of points of any $\infty$-scheme would be the same as the functor of points of the associated DM $\infty$-stack, which is an étale sheaf.