Module Categories As Spans
Hao Xu
TL;DR
This work develops a comprehensive framework linking modules over algebras in semistrict monoidal 2-categories to spans of algebras. It introduces a delooping-based enrichment viewpoint, defines A-modules as enriched functors from the delooping $f BA$ to a chosen enriched 2-category, and constructs a normalized lax 3-functor from the 2-category of A-modules to 2-spans of algebras under A. The core achievement is a systematic dictionary that translates module-theoretic data (objects, 1- and 2-morphisms) into spans of endomorphism algebras, with explicit laxator coherence. The framework specializes to classical settings (e.g., module categories over monoidal categories) and yields concrete spans in numerous monoidal 2-categories (such as $ extbf{Cat}$, $ extbf{2Vect}$, and braided/symmetric variants), while outlining extensions to $ extbf{MCat}$ and $ extbf{BrCat}$ and connections to concepts like central module monoidal categories and bulk-boundary correspondences in field theories.
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.