Tensor functors between categories of quasi-coherent sheaves – arXiv:1202.5147

This paper by Martin Brandenburg and Alexandru Chirvasitu looks interesting. There is a connection to a categorified algebraic geometry. From the introduction:

Apart from being interesting in its own right, this question arises naturally as part of the discussion on “2-algebraic geometry” in [JC]. In that paper, a notion of commutative 2-ring is introduced. These are symmetric monoidal categories satisfying some extra technical conditions; the important thing for us is that categories of the form Qcoh(X) are examples of commutative 2-rings, and one would like to conclude that X |→ Qcoh(X) is fully faithful, and hence one can recover 1-algebraic geometry as affine 2-algebraic geometry

ArXiv:1202.5147

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.