Categorification of modules and construction of schemes

TL;DR

This paper develops a framework for algebraic geometry over symmetric monoidal categories using categorification, introducing schemes over a datum involving an actegory and a spectral topology, extending classical geometric concepts.

Contribution

It introduces a new approach to algebraic geometry over symmetric monoidal categories via categorification and defines schemes over a specified datum, unifying existing theories.

Findings

Defined schemes over a symmetric monoidal category and an actegory.

Introduced the spectral M-topology with fpqc M-coverings.

Showed categories of schemes are closed under pullbacks and coproducts.

Abstract

We use categorification of monoid actions to study algebraic geometry over symmetric monoidal categories. This brings together the relative algebraic geometry over symmetric monoidal categories developed by To\"{e}n and Vaqui\'{e}, along with the theory of actegories 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 an actegory 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 provide a description of schemes over $(\mathcal C,\mathcal M)$ in terms of quotients of disjoint unions of affine schemes over a certain equivalence relation. These categories of schemes are closed under pullbacks and coproducts, and are equipped with change of base functors induced by symmetric monoidal adjuctions accompanied by lax $\mathcal C$-linear functors.