nLab
Burnside category

Contents

Contents

Idea

By the Burnside category of a finite group GG one means either

  1. the category of correspondences in finite G-sets (Prop. below)

  2. or an abelianization of that

    1. either by group completion of its monoidal hom-groupoids to an additive 1-category (Def. below);

    2. or by forming the K-theory of a permutative category of the hom-groupoids to a spectra-enriched (∞,1)-category (Def. below).

The plain Burnside ring is the endomorphism ring of the terminal G-set in the additive version of the category (Example below).

The functors/additive functors/enriched (∞,1)-functors from the Burnside category to, in particular, the category of abelian groups/(∞,1)-category of spectra are called Mackey functors.

A functor out of the Burnside category is called a Mackey functor, specifically if it is an Ab-enriched functor to Ab out of the additive category-version of the Burnside category. A spectra-enriched functor out of the spectral Burnside category to Spectra is called a spectral Mackey functor, for emphasis. These spectral Mackey functors on the Burnside category are equivalent to genuine G-spectra in equivariant stable homotopy theory.

Definition

There are various incarnations of the Burnside category as enriched categories, in varying degree of sophistication of the enriching “cosmos”. They are all induced from the canonical structure of permutative categories on the correspondences between two fixed finite G-sets (made explicit as Prop. below). This yields the Burnside category as

  1. enriched in permutative categories;

  2. enriched in abelian groups;

  3. enriched in spectra.

Proposition

(the permutative category of finite G-sets)

For GG be a finite group, write GFinSet skG FinSet_{sk} for the skeleton of the category of finite G-sets.

Its objects may be identified with pairs (n,ρ)(n,\rho) consisting of a natural number nn, reflecting the finite set {1,2,,n}\{1,2, \cdots, n\}, and a group homomorphism ρ:GS n=Aut({1,2,,n})\rho \;\colon\; G \longrightarrow S_n = Aut(\{1,2, \cdots, n\}) from GG to the symmetric group on nn elements, reflecting the automorphism group of that finite set.

Its morphisms (n 1,ρ 1)f(n 2,ρ 2)(n_1,\rho_1) \overset{ f }{\longrightarrow} (n_2, \rho_2) are functions f:{1,2,,n 1}{1,2,,n 2}f \;\colon\; \{1, 2, \cdots, n_1\} \longrightarrow \{1,2, \cdots, n_2\} that intertwine ρ 1\rho_1 and ρ 2\rho_2.

The coproduct \sqcup of G-sets (disjoint union) makes this skeleton a permutative category (GFinSet sk,)\big( G FinSet_{sk}, \sqcup \big).

In the same way its slice category over any object XGFinSet skX \in G FinSet_{sk} becomes a permutative category (GFinSet sk/ X,)\big( G FinSet_{sk}/_{X}, \sqcup \big) under disjoint union of domains.

Similarly, the Cartesian product of finite GG-sets can be restricted to these skeleta to produce bipermutative categories (GFinSet sk/ X,,×)\big( G FinSet_{sk}/_{X}, \sqcup, \times \big).

Finally, restriction to isomorphisms (passage to cores) yields the bipermutative groupoids (Core(GFinSet sk)/ X,,×)\big( Core(G FinSet_{sk})/_{X}, \sqcup, \times \big).

(e.g. Guillou-May 11, Def. 1.3, Bohmann-Osorno 14, Def. 1.4)

As enriched in permutative categories

Definition

(PermCat-enriched Burnside category)

The (2,1)-category of correspondences Corr(GFinSet)Corr(G FinSet) is equivalent to the (2,1)-category whose objects are the finite G-sets and whose hom-categories are the permutative cores of skeleta of slice categories of G-sets from Def. , over the Cartesian product of source and domain G-sets:

Corr(GFinSet)(S 1,S 2)Core(GFinSet sk/ S 1×S 2). Corr(G FinSet)\big( S_1, S_2 \big) \;\simeq\; Core(G FinSet_{sk}/_{S_1 \times S_2}) \,.

This locally skeletal (2,1)-category is the PermCat-enriched Burnside category GBurn pcG Burn_{pc}_

It may be regarded as an enriched category over the multicategory PermCat of permutative categories (a PC\mathbf{PC}-cateory in the sense of Guillout 10).

(Guillou-May 11, Def. 1.6, Bohmann-Osorno 14, Def. 7.1, 7.2)

As enriched in abelian groups

Definition

(additive Burnside category)

The additive Burnside category GBurn adG Burn_{ad} of GG is the additive category obtained from the PermCat-enriched Burnside category GBurn pcG Burn_{pc} (Def. ) under replacing each hom-permutative category Core(GFinSet sk/ S 1×S 2)Core(G FinSet_{sk}/_{S_1 \times S_2}) with its Grothendieck group, hence with the abelian group which is the group completion

K:PermCatiso classesCMonKAb K \;\colon\; PermCat \overset{\text{iso classes}}{\longrightarrow} CMon \overset{K}{\longrightarrow} Ab

of the commutative monoid of isomorphism classes of objects in Core(GFinSet sk/ S 1×S 2)Core(G FinSet_{sk}/_{S_1 \times S_2}), under disjoint union:

GBurn abK GBurn pc. G Burn_{ab} \;\coloneqq\; K_{\bullet} G Burn_{pc} \,.
Example

(Burnside ring is endomorphism ring of additive Burnside category)

The endomorphism ring of the terminal G-set (the point *\ast equipped with the, necessarily, trivial action) in the additive Burnside category (Def. ) is the Burnside ring A(G)A(G):

End GBurn ad(*,*)A(G) End_{G Burn_{ad}}(\ast, \ast) \;\simeq\; A(G)

As enriched in spectra

Definition

(Spectra-enriched Burnside category)

Since construction of K-theory spectra of permutative categories applies hom-object-wise to PermCat-enriched categories (this Prop.)

𝕂 :PermCatCatSpectraCat \mathbb{K}_\bullet \;\colon\; PermCat Cat \longrightarrow Spectra Cat

the image of the PermCat-enriched Burnside category GG \mathcal{E} from Def. under forming hom-object-wise the K-theory spectra of permutative categories yields a Spectra-enriched category

GBurn sp𝕂 GBurn pc. G Burn_{sp} \;\coloneqq\; \mathbb{K}_\bullet G Burn_{pc} \,.

This is called the spectral Burnside category.

(Guillou-May 11, Def. 1.12, Bohmann-Osorno 14, Def. 7.3)

Properties

Relation to GG-Spectra

Proposition

(G-spectra are spectral presheaves on the spectral Burnside category)

There is a zig-zag of Quillen equivalences

[GBurn sp,Spectra] Qu zigzagGSpectra [ G Burn_{sp}, Spectra ] \;\simeq_{Qu}^{zigzag}\; G Spectra

between the model category of Spectra-enriched presheaves over the spectral Burnside category from Def. (“spectral Mackey functors”) and that of genuine G-spectra.

This equivalence is such that the spectral Mackey functor corresponding to a fibrant G-spectrum EE assigns to the transitive G-set G/HG/H the fixed point spectrum E HE^H:

G/HE(G/H)[Σ G G/H,E] GE H. G/H \;\mapsto\; E(G/H) \;\coloneqq\; [\Sigma^\infty_G G/H, E]^G \;\simeq\; E^H \,.

(GuillouMay 11, theorem 1.13, corollary 1.14, remark 2.5).

References

Lecture notes include:

The spectrally-enriched version and its role in the equivalent description of G-spectra via spectral Mackey functors is due to

A beautified review is given in

making use of

  • Bertrand Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ., 24:No. 20, 564–579, 2010 (TAC)

Last revised on January 2, 2019 at 09:29:26. See the history of this page for a list of all contributions to it.