The additive completion of the biset category

TL;DR

This paper constructs an additive, symmetric monoidal category from bisets over finite groups, showing its equivalence to the additive completion of the biset category and relating biset functors to functors from this new category.

Contribution

It introduces a new category $\\mathcal{C}_R$ that completes the biset category additively and demonstrates its equivalence to biset functor categories, providing a new framework for studying biset functors.

Findings

$\\mathcal{C}_R$ is additive, symmetric monoidal, and self-dual.

$\\mathcal{C}_R$ is equivalent to the additive completion of the biset category.

Biset functors over $R$ are equivalent to $R$-linear functors from $\mathcal{C}_R$.

Abstract

Let $R$ be a commutative unital ring. We construct a category $\mathcal{C}_R$ of fractions $X/G$, where $G$ is a finite group and $X$ is a finite $G$-set, and with morphisms given by $R$-linear combinations of spans of bisets. This category is an additive, symmetric monoidal and self-dual category, with a Krull-Schmidt decomposition for objects. We show that $\mathcal{C}_R$ is equivalent to the additive completion of the biset category and that the category of biset functors over $R$ is equivalent to the category of $R$-linear functors from $\mathcal{C}_R$ to $R$-Mod. We also show that the restriction of one of these functors to a certain subcategory of $\mathcal{C}_R$ is a fused Mackey functor.