Abstract

The classical biset category (with a modification) first arose in a description of the morphisms between classifying spaces of finite groups, as a consequence of the Segal Conjecture. Biset functors are linear functors from this category to abelian groups. They are closely related to Mackey functors, being sometimes called globally defined Mackey functors. The classical theory has since been extended in several directions: one is that the biset category can now be taken to have as its objects all finite categories, rather than just all finite groups, so that biset functors are now defined on arbitrary finite categories. A key role is played by the Burnside ring of the finite category, for which the definition is new when the category is not a group. The homology and cohomology of simplicial complexes can both be regarded as biset functors, and this provides a uniform setting for the construction of transfer maps.  Another direction of development is that both the biset category and the category of biset functors are monoidal, and the biset category is, in fact, rigid. I give an overview of key aspects of this theory.