Burnside category

Category theory

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

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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 <- U -> Y {\displaystyle X\leftarrow U\rightarrow Y} where two spans  X <- U -> Y {\displaystyle X\leftarrow U\rightarrow Y} and  X <- W -> Y {\displaystyle 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 ) {\displaystyle A(G)(X,Y)} the group completion of that monoid. Taking pullbacks induces natural maps  A ( G ) ( X , Y ) × A ( G ) ( Y , Z ) -> A ( G ) ( X , Z ) {\displaystyle 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 ) {\displaystyle A(G)(X,Y)} .

## Properties

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* 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.[1]

## Mackey functors

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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

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

* Guillou, Bertrand; May, J.P. (2011). "Models of G-spectra as presheaves of spectra". arXiv:1110.3571 [math.AT].
* Barwick, Clark (2014). "Spectral Mackey functors and equivariant algebraic K-theory (I)". arXiv:1404.0108 [math.AT].