John Carlos Baez on Nostr: A conversation with James Dolan and Chris Grossack: 2023-12-15 ...
A conversation with James Dolan and Chris Grossack:
2023-12-15
https://www.youtube.com/watch?v=TbRrYwMgkEY
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.
(4/n)
2023-12-15
https://www.youtube.com/watch?v=TbRrYwMgkEY
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.
(4/n)