higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
An atlas is a compatible collection of coordinate charts.
In full generality, for $\mathcal{G}$ a pregeometry and $X \in Sh_{(\infty,1)}(\mathcal{G})$ an object in the (∞,1)-sheaf (∞,1)-topos, an atlas for $X$ is a collection of suitable morphisms (open maps) $\{U_i \to X\}$ with $U_i \in \mathcal{G} \hookrightarrow Sh_{(\infty,1)}(\mathcal{G})$, such that the morphism out of the coproduct
is an effective epimorphism.
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