nLab 2-algebraic geometry

Idea

Where in algebraic geometry one considers spaces which are formally dual to commutative rings, in “2-algebraic geometry” one considers spaces formally dual to 2-rigs, namely to certain tensor categories or more generally to tensor (∞,1)-categories.

Curiously, the more general idea of regarding certain abelian categories and certain linear A-∞ categories (stable (∞,1)-categories) as analogs of categories of quasicoherent sheaves on some space (i.e. ignoring the monoidal structure) is much older, this is the program of derived noncommutative geometry, see also at derived algebraic geometry – Relation to noncommutative geometry.

The systematic use of the tensor product structure here goes back to (Balmer 02) and the concept of the spectrum of a tensor triangulated category. In (Lurie) the concept of “2-affine scheme” is studied more systematically. Then (CJF 11) introduce the terminology of “2-algebraic geometry”.

Related concepts

• spectrum of a tensor triangulated category

• spectrum of a symmetric monoidal (∞,1)-category

References

• Paul Balmer, Presheaves of triangulated categories and reconstruction of schemes, Mathematische Annalen 324:3 (2002), 557-580 dvi, pdf ps; The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588:149–168, 2005 dvi pdf ps; Spectra, spectra, spectra - Tensor triangular spectra versus Zariski spectra of endomorphism rings, Alg. and Geom. Topology 10:3 (2010) 1521-1563 dvi pdf ps

• Jacob Lurie, Tannaka duality for geometric stacks.

• Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. (DOI, arXiv:1105.3104)

• Martin Brandenburg, Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)