2-torials / Categorical algebraic geometry
Further reading
- Bertrand Toën, Michel Vaquié, “Under Spec Z”. arXiv:math/0509684
- Bertrand Toen, Gabriele Vezzosi, “Homotopical Algebraic Geometry II: geometric stacks and applications”. arXiv:math/0404373
Tim Hosgood
Jason Brown
July 28, 2026
There are many approaches to algebraic geometry, and many different ways to arrive at the subject. Here we’re going to take an incredibly specific and biased approach: what if you already love 2-categories and want to be able to say the phrase “fpqc sheaf” as quickly as possible, but not too quickly? In this series of exercises we will build towards an understanding of relative algebraic geometry, which allows us to work in arbitrary (nice) symmetric monoidal categories.
Functor of points, affine scheme, quasi-coherent sheaf, Grothendieck pseudofunctor, pre-topology, faithfully flat topology, algebra over a commutative monoid, sheaf, affine scheme (again).
Quotients of rings, Yoneda embedding, 2-limits, symmetric monoidal categories (cosmoi).
Further reading