Tensor functors between categories of quasi-coherent sheaves

TL;DR

This paper establishes an equivalence between geometric morphisms and tensor functors between categories of quasi-coherent sheaves, extending to generalized schemes over F_1, thus showing algebraic geometry's 2-affineness.

Contribution

It improves Lurie's result by proving an equivalence between morphisms and tensor functors for quasi-coherent sheaves, including generalized schemes over F_1.

Findings

Equivalence between morphisms Y --> X and cocontinuous tensor functors Qcoh(X) --> Qcoh(Y).

Extension of the equivalence to generalized schemes over F_1.

Demonstration that algebraic geometry is 2-affine.

Abstract

For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction implements an equivalence between the discrete category of morphisms Y --> X and the category of cocontinuous tensor functors Qcoh(X) --> Qcoh(Y). This is an improvement of a result by Lurie and may be interpreted as the statement that algebraic geometry is 2-affine. Moreover, we prove the analogous version of this result for Durov's notion of generalized schemes over F_1.