The additive completion of the biset category

Let R be a commutative unital ring. We construct a category CR of fractionsX/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 decom...

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Published inJournal of pure and applied algebra Vol. 222; no. 2; pp. 297 - 315
Main Authors Ibarra, Jesús, Raggi-Cárdenas, Alberto G., Romero, Nadia
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.02.2018
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Summary:Let R be a commutative unital ring. We construct a category CR of fractionsX/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 CR 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 CR to R-Mod. We also show that the restriction of one of these functors to a certain subcategory of CR is a fused Mackey functor.
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2017.04.003