The additive completion of the biset category

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.