higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
For a ringed topos $(\mathcal{X}, \mathcal{O})$ the ring object $\mathcal{O} \in \mathcal{X}$ is called the structure sheaf.
More generally, for $\mathcal{G}$ a geometry (for structured (∞,1)-toposes), a structured (∞,1)-topos
is an (∞,1)-topos equipped with a $\mathcal{G}$-valued structure sheaf presented by the finite-limits-preserving and cover-preserving (∞,1)-functor $\mathcal{O}$.