# Measuring Comodules and Enrichment

## TL;DR

This paper generalizes the concept of universal measuring comonoids to modules and comodules within braided monoidal categories, establishing foundational theorems and exploring applications like higher derivations and non-commutative Hasse-Schmidt algebra.

## Contribution

It extends the theory of universal measuring comonoids to arbitrary braided monoidal categories and proves key representability and adjoint functor theorems.

## Key findings

- Universal measuring comodule Q(M,N) exists in braided monoidal categories.
- Modules are enriched in comodules within this framework.
- Framework applied to higher derivations and non-commutative Hasse-Schmidt algebra.

## 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.