On Enriched Fibrations

TL;DR

This paper introduces enriched fibrations, a new categorical structure that generalizes fibrations with enrichment, and provides methods to construct them from monoidal category actions, with applications to modules and comodules.

Contribution

It defines enriched fibrations and offers a construction approach from monoidal category actions with parameterized adjoints.

Findings

Defines the concept of enriched fibrations.

Provides a construction method from monoidal actions.

Applies to modules over algebras and comodules over coalgebras.

Abstract

We introduce the notion of an enriched fibration, i.e. a fibration whose total category and base category are enriched in those of a monoidal fibration in an appropriate way. Furthermore, we provide a way to obtain such a structure, starting from actions of monoidal categories with parameterized adjoints. The motivating goal is to capture certain example cases, like the fibration of modules over algebras enriched in the opfibration of comodules over coalgebras.