higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
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Structured Spaces
which develops the generalization of the notion of a ringed topos from topos theory to (∞,1)-topos theory and formulates basic notions of geometry in this context (“higher geometry”).
Precursor in 1-category theory include
Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).
The theory is general, but the focus of the examples and applications is on derived algebraic geometry/E-∞ geometry.
A survey is at A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves in the section notions of space.