Semigroup actions on sets and the Burnside ring

TL;DR

This paper explores the structure of categories of semigroup actions, introduces homotopical frameworks, and constructs a Burnside ring for monoids, generalizing classical group-based concepts.

Contribution

It develops enlarged categories of semigroup actions with homotopical structures and defines a Burnside ring for monoids, extending classical group theory results.

Findings

Enlarged categories have two idempotent endofunctors.

Homotopical categories are equivalent to usual actions up to homotopy.

Burnside ring of a monoid generalizes the group case and relates to the Grothendieck group.

Abstract

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gr\"othendieck group.