Commutative algebras and representations of the category of finite sets

TL;DR

This paper establishes a characterization of finite-dimensional commutative algebras over an algebraically closed field through their associated representations of the category of finite sets and surjective maps.

Contribution

It proves that isomorphism classes of such algebras correspond exactly to isomorphism classes of their representations of the finite sets category.

Findings

Finite-dimensional commutative algebras are characterized by their representations.

Isomorphism of algebras corresponds to isomorphism of their representations.

Provides a categorical perspective on algebra isomorphism.

Abstract

We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.