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 $X\leftarrow U \rightarrow Y$ where two spans $X\leftarrow U \rightarrow Y$ and $X\leftarrow W \rightarrow Y$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 $A(G)(X,Y)$ the group completion of that monoid. Taking pullbacks induces natural maps $A(G)(X,Y)\times A(G)(Y,Z)\rightarrow A(G)(X,Z)$.

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 $A(G)(X,Y)$.

Properties

* ''A''(''G'') is an additive category with direct sums given by the disjoint union of ''G''-sets and zero object given by the empty ''G''-set;
* The product of two ''G''-sets induces a symmetric monoidal structure on ''A''(''G'');
* The endomorphism ring of the point (that is the ''G''-set with only one element) is the Burnside ring of ''G'';
* ''A''(''G'') is equivalent to the full subcategory of the homotopy category of genuine ''G''-spectra spanned by the suspension spectra of finite ''G''-sets.

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.
* To every ''G''-representation ''V'' we can associate a Mackey functor in vector spaces sending every finite ''G''-set ''U'' to the vector space of ''G''-equivariant maps from ''U'' to ''V''.
* The homotopy groups of a genuine ''G''-spectrum form a Mackey functor. In fact genuine ''G''-spectra can be seen as additive functor on an appropriately higher categorical version of the Burnside category.

References

*
* {{cite arXiv , last=Barwick , first=Clark , title=Spectral Mackey functors and equivariant algebraic K-theory (I) , eprint=1404.0108

Category theory
Group theory