# 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[edit]

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 where two spans and 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 .

## Properties[edit]

*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.- The Burnside category is self-dual.
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## Mackey functors[edit]

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[edit]

**^**Dugger, Daniel (2022). "GYSIN FUNCTORS, CORRESPONDENCES, AND THE GROTHENDIECK-WITT CATEGORY" (PDF).*Theory and Application of Categories*.**38**(6): 158.