Burnside category

In category theory and homotopy theory the Burnside category of a finite group "G" is a category whose objects are finite "G"-sets and whose morphisms are (equivalence classes of) spans of "G"-equivariant maps. It is a categorification of the Burnside ring of "G".
Definitions.
Let "G" be a finite group (in fact everything will work verbatim for a profinite group). Then for any two finite "G"-sets "X" and "Y" we can define an equivalence relation among spans of "G"-sets of the form formula_1 where two spans formula_1 and formula_3are equivalent if and only if there is a "G"-equivariant bijection of "U" and "W" commuting with the projection maps to "X" and "Y". This set of equivalence classes form naturally a monoid under disjoint union; we indicate with formula_4 the group completion of that monoid. Taking pullbacks induces natural maps formula_5.
Finally we can define the Burnside category "A(G)" of "G" as the category whose objects are finite "G"-sets and the morphisms spaces are the groups formula_4.
Mackey functors.
If "C" is an additive category, then a "C"-valued Mackey functor is an additive functor from "A(G)" to "C". Mackey functors are important in representation theory and stable equivariant homotopy theory.