higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A locally ringed topos is a locally algebra-ed topos for the theory of local rings.
A ringed topos $(X, \mathcal{O}_X)$ with enough points (such as the sheaf topos over a topological space) is a locally ringed topos if all stalks $\mathcal{O}_X(x)$ are local rings.
This is a special case of the following equivalent definitions:
A locally ringed topos is a topos equipped with a commutative ring object (see ringed topos) that in addition satisfies the axioms
(note these are axioms for a geometric theory, interpreted according to Kripke-Joyal semantics in a topos).
A ringed topos $(X, \mathcal{O}_X)$ is a locally algebra-ed topos for the theory of local rings:
a topos $X$
equipped with a geometric morphism
into the Zariski topos, the classifying topos for the theory of local rings.
This is for instance in ([Johnstone]) and in (Lurie, remark 2.5.11)
ringed topos, locally ringed topos
Section VIII.6 of
Section abc of
Section 2.5 of
Section 14.33 of