Categorification of modules and construction of schemes
Abhishek Banerjee, Subhajit Das, Surjeet Kour
TL;DR
The paper develops a framework for algebraic geometry over symmetric monoidal categories by categorifying module actions through a datum $(\mathcal{C},\mathcal{M})$ where $\mathcal{M}$ is a $\mathcal{C}$-actegory. It defines affine schemes $Aff_{\mathcal{C}} = Comm(\mathcal{C})^{op}$, introduces the spectral $\mathcal{M}$-topology via fpqc $\mathcal{M}$-coverings, and defines $\mathcal{M}$-schemes as sheaves with affine Zariski $\mathcal{M}$-covers, along with a base-change formalism for changing base categories $\mathcal{C} \to \mathcal{D}$. The work establishes stability properties, a quotient description of schemes, and a robust change-of-base theory, then explores many examples illustrating subcanonical topologies in various categorical settings. Finally, it extends Connes–Consani’s gluing by adjoin ing $CMon_0$ to obtain a unified notion of scheme over a composite datum, enabling a potential synthesis of categorical and monoid-based geometries. Overall, the results provide a comprehensive, base-change-friendly framework for categorified algebraic geometry over symmetric monoidal contexts.
Abstract
We use categorification of module structures to study algebraic geometry over symmetric monoidal categories. This brings together the relative algebraic geometry over symmetric monoidal categories developed by Toën and Vaquié, along with the theory of module categories over monoidal categories. We obtain schemes over a datum $(\mathcal C,\mathcal M)$, where $(\mathcal C,\otimes,1)$ is a symmetric monoidal category and $\mathcal M$ is a module category over $\mathcal C$. One of our main tools is using the datum $(\mathcal C,\mathcal M)$ to give a Grothendieck topology on the category of affine schemes over $(\mathcal C,\otimes,1)$ that we call the ``spectral $\mathcal M$-topology.'' This consists of ``fpqc $\mathcal M$-coverings'' with certain special properties. We also give a counterpart for a construction of Connes and Consani by presenting a notion of scheme over a composite datum consisting of a $\mathcal C$-module category $\mathcal M$ and the category of commutative monoids with an absorbing element.