Categorification of modules and construction of schemes

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 To\"{e}n and Vaqui\'{e}, 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 offpqc $\mathcal M$-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.