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”).
A 1-categorical precursor would be
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.