higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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”.
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
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)
Last revised on August 1, 2021 at 03:05:51. See the history of this page for a list of all contributions to it.