Measuring Comodules and Enrichment
Martin Hyland, Ignacio Lopez Franco, Christina Vasilakopoulou
TL;DR
This work develops a broad, category-theoretic framework unifying measuring constructions for modules and comodules in braided monoidal categories. By establishing representability results for presheaves on opfibrations and an adjoint-functor theorem in that setting, it proves that the global category of modules is enriched in the global category of comodules, with the universal measuring comodule $Q(M,N)$ and the universal measuring coalgebra $P(A,B)$ as central objects. The paper further connects these universal measurings to derivations and higher derivations, introducing the non-commutative Hasse–Schmidt algebra via Sweedler-type constructions and showing how coinvariants and base-change behave under this enrichment. The resulting enriched, tensored/cotensored structure generalizes classical Sweedler duals and provides a robust toolkit for analyzing derivations and jet-like structures in a braided-m monoidal context, with potential applications to higher-derivation theory and non-commutative differential geometry.
Abstract
This paper extends the theory of universal measuring comonoids to modules and comodules in braided monoidal categories. We generalise the universal measuring comodule Q(M,N), originally introduced for modules over k-algebras when k is a field, to arbitrary braided monoidal categories. In order to establish its existence, we prove a representability theorem for presheaves on opfibred categories and an adjoint functor theorem for opfibred functors. The global categories of modules and comodules, fibred and opfibred over monoids and comonoids respectively, are shown to exhibit an enrichment of modules in comodules. Additionally, we use our framework to study higher derivations of algebras and modules, defining along the way the non-commutative Hasse-Schmidt algebra.