Construction of coend and the reconstruction theorem of bialgebras

TL;DR

This paper provides a categorical construction of coend and end for a functor from a linear abelian category to vector spaces, and uses this to give a concrete reconstruction theorem for bialgebras, especially when the functor is tensorial.

Contribution

It introduces a constructive approach to describe end and coend in linear categories and applies this to establish a new reconstruction theorem for bialgebras.

Findings

Explicit description of end(F) and coend(F) in linear categories

Concrete bialgebra structure of coend(F) for tensor functors

Reconstruction theorem for bialgebras using categorical constructions

Abstract

Assume $k$ is a field and let $F:C\rightarrow Vect_{k}$ be a small $k$-linear functor from a $k$-linear abelian category $C$ to the category of vector spaces over the field $k$, the purpose of this note is to use a little knowledge of linear algebra and category to give the description of $end(F)$ and $coend(F)$, and then we give the reconstruction theorem of bialgebras by using this description. We use a constructive approach to understand $end(F),coend(F)$ and we describe the bialgebra structure of $coend(F)$ concretely when $F$ is a tensor functor.