Coends and the tensor product of $\mathcal{C}$-modules

TL;DR

This paper explores the relationships between Kan extensions, coends, and tensor products of functors, providing detailed proofs and formulas to compute these concepts, especially focusing on $ ext{C}$-modules.

Contribution

It introduces a detailed study of Kan extensions and coends, and demonstrates how to compute tensor products of $ ext{C}$-modules using these concepts.

Findings

Provides a detailed formula for computing Kan extensions via coends.

Shows how the tensor product of $ ext{C}$-modules can be represented as a specific case of Kan extensions.

Establishes connections between coends, adjoint functors, and tensor products in category theory.

Abstract

We give an introduction to the concept of Kan extensions, and study its relation with the notions of coend and adjoint functors. We state and prove in detail a well known formula to compute Kan extensions by using coends: a certain colimit related to the concept of copower. Finally, we study the tensor product of functors, and its relation with Kan extensions, in order to represent the tensor product of $\mathcal{C}$-modules as a particular case.