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. 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
are 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
the group completion of that monoid.
Taking pullbacks induces natural maps
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
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.