Baez, Dolan and Grossack, 2023-12-15

Categorifying various attitudes to rings, or rigs, to get corresponding attitudes to 2-rigs. A commutative algebraist studies commutative rings while an algebraic geometer might work in the opposite category and think of them as affine schemes. The algebraic side is more 'syntactical' while the geometric side is more 'semantic'. You might think the geometric interpretation of a 2-rig is typically some sort of 'categorified affine scheme', but that's not always true! For example, if you take the 2-rig of modules of a commutative ring R, its spectrum is the same as that of R.
However, most 2-rigs aren't module categories of rings. Take a quasiprojective variety X and look at the 2-rig of quasicoherent sheaves on it, QCoh(X). When X is an affine variety QCoh(X) is equivalent to the 2-rig of modules of a ring, namely the ring R with Spec(R) ≅ X. But when X is a projective variety this is not true.
The free commutative ring on one generator is ℤ[x]. If we think of this as a space it's the line, which is an affine scheme. Similarly, the 2-rig of modules of ℤ[x] is the 2-rig of quasicoherent sheaves on the line, which is an affine scheme.
On the other hand, the free 2-rig on a line object is the 2-rig of ℤ-graded vector spaces, which is equivalent to the 2-rig of algebraic representations of the affine group scheme GL(1), or comodules of ℤ[x]. If we think of this 2-rig as a kind of 'space' it's the algebraic stack BGL(1).
Next consider the free 2-rig on a line object equipped with a line object equipped with a monomorphism to I⊕I, where I is the tensor unit. The corresponding space is the projective line P¹. For details see:
Martin Brandenburg, Tensor categorical foundations of algebraic geometry, arxiv.org/abs/....
Also see Deligne's work on tensor categories, discussed here:
ncatlab.org/nl...
which shows that a large class of 2-rigs are categories of representations of supergroups. A more general attitude toward the 'Tannakian philosophy' also considers supergroupoids (coming from 'non-neutral' Tannakian categories, as defined in the above nLab article).
For more on this whole series of conversations, go here:
math.ucr.edu/h...
For more on this whole series of conversations, go here:
math.ucr.edu/h...