higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
… derived algebraic geometry … higher algebra …generalized scheme…
Let $k$ be a commutative ring.
A derived scheme (over $k$) is a generalized scheme in the sense of locally affine $\mathcal{G}$-structured (infinity,1)-topos for $\mathcal{G} = \mathcal{G}_{Zar}(k)$ the Zariski geometry (for structured (infinity,1)-toposes).
A 0-trucated and 0-localic derived scheme is precisely an ordinary scheme.
More precisely:
Let $Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}(k)) \subset Sch(\mathcal{G}_{Zar}(k))$ be the full subcategory of all derived schemes on the 0-trucated and 0-localic ones. This is canonically equivalent to the ordinary category $Sch(k)$ of schemes over $k$:
For more comments on this see also
Notice that for generalized schemes the Zariski geometry (for structured (infinity,1)-toposes) $\mathcal{G}_{Zar}(k)$ is not interchangeable with the étale (∞,1)-site $\mathcal{G}_{et}(k)$. Instead $\mathcal{G}_{et}(k)$-generalized schemes are derived Deligne-Mumford stacks.
section 4.2 in