Module Categories As Spans

Abstract

We establish a correspondence between modules and spans of algebras within a general monoidal 2-category $\mathfrak{C}$. Specifically, for an algebra $A$ in $\mathfrak{C}$, we construct a normalized lax 3-functor from the 2-category of $A$-modules to the 3-category of 2-spans of algebras in $\mathfrak{C}$ under $A$. This framework unifies and generalizes the realization of module functors and module natural transformations as spans of monoidal functors. We demonstrate the utility of this theory by recovering the realization of module objects in several familiar 2-categories and discuss its extension to the 2-categories $\mathbf{MCat}$ and $\mathbf{BrCat}$. In these cases, module objects correspond to central module monoidal categories over a braided monoidal category and central braided monoidal categories over a symmetric monoidal category, respectively.