This is just a quick thought that I’m putting up here so that I can link to it.
Define \mathbb{R}-\mathsf{Idl} to be the following double category.
- The objects are finite sets
- A vertical map A \to A' is a function A' \to F(A), where F(A) is the free \mathbb{R}-algebra on the finite set A.
- A horizontal map A \to B is an ideal I \subseteq F(A+B)
- A 2-cell
exists if and only if (f+g)(I') \subset I.
We compose ideals in the following manner. Suppose that I \subset F(A+B) and J \subset F(B+C). Then we can make I \oplus J \subset F(A+B) \oplus F(B+C) \cong F(A+B+B+C). Now, there are natural Kleisli maps A + B + B + C \to A + B + C \leftarrow A + C. We can then pull back and push forward I \oplus J along these maps to make I ; J \subset F(A + C).
I conjecture that this is an equipment because a map f \colon A \to A' can be turned into an ideal \mathrm{graph}(f) generated by elements of the form x - f(x) for x \in A'. Note that \mathrm{Spec}(F(A+A')/\mathrm{graph}(f)) \cong \mathrm{graph}(\mathrm{Spec}(f)), where \mathrm{Spec}(f) \colon \mathrm{Spec}(A) \to \mathrm{Spec}(A').
It is Cartesian because + is the Cartesian product in the opposite of the Kleisli category of F.
The intuition for this is that it’s a symbolic version of the double category \mathsf{Rel}. Additionally, this lets us talk about algebraic varieties in terms of generators and relations explicitly, because the generators and relations are split up into the objects and proarrows respectively. The horizontal category here is essentially equivalent to the category of finitely presented \mathbb{R}-algebras.