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2-algebraic geometry

Contents

Context

Higher geometry

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

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”.

References

Created on October 9, 2014 at 20:46:49. See the history of this page for a list of all contributions to it.