nLab Structured Spaces

This entry provides hyperlinks for central keywords of the text

• Jacob Lurie,

  Structured Spaces

  (arXiv:0905.0459, pdf)

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) (doi:10.1007/978-3-662-59155-0)

• Cole's theory of spectrum

The theory is general, but the focus of the examples and applications is on derived algebraic geometry/E-∞ geometry.

Apparently a version of this is to eventually appear as:

• Jacob Lurie, part I Fundamentals of Spectral Algebraic Geometry, of: Spectral Algebraic Geometry

Survey

A survey is at A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves in the section notions of space.

Contents

• higher geometry

• structured (∞,1)-topos

  • geometry (for structured (∞,1)-toposes)

  • structure sheaf

  • generalized scheme

    • derived algebraic geometry

      • derived scheme

      • derived Deligne-Mumford stack

    • derived smooth manifold

category: reference