Sweedler theory for double categories

TL;DR

This paper develops a double categorical framework to enrich algebraic structures like monads and modules, generalizing classical concepts and applying to V-matrices, extending Sweedler's measuring coalgebras.

Contribution

It introduces a double categorical approach to enrich monads, comonads, modules, and comodules, generalizing classical algebraic notions within this setting.

Findings

Established enrichments of monads in comonads and modules in comodules.

Connected the framework to V-matrices and generalized Sweedler's measuring coalgebras.

Demonstrated the enrichment of modules over monoids in a double categorical context.

Abstract

In this work, we establish certain enrichments of dual algebraic structures in the setting of monoidal double categories. In more detail, we obtain a tensored and cotensored enrichment of monads in comonads, as well as a tensored and cotensored enrichment of modules in comodules, under very general conditions on the surrounding double category. These include `monoidal closedness' and `local presentability', leading classical notions which are here introduced in the double categorical context. Furthermore, we show that in this setting, the actual fibration of modules over monads is itself enriched in the opfibration of comodules over comonads. Applying this abstract double categorical framework to the setting of V-matrices, one directly obtains a many-object generalization of the known enrichment of modules over monoids in comodules over comonoids in a monoidal category V, which was originally induced by Sweedler's measuring k-coalgebras in vector spaces. In the present setting the result involves V-enriched modules of V-categories and V-enriched comodules of V-cocategories.