nLab Burnside category

Idea

By the Burnside category of a finite group $G$ 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 $G$ be a finite group, write $G FinSet_{sk}$ for the skeleton of the category of finite G-sets.

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

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

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

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

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

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

(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 $Corr(G FinSet)$ 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:

$$Corr(G FinSet)\big( S_1, S_2 \big) \;\simeq\; Core(G FinSet_{sk}/_{S_1 \times S_2}) \,.$$

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

It may be regarded as an enriched category over the multicategory PermCat of permutative categories (a $\mathbf{PC}$-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 $G Burn_{ad}$ of $G$ is the additive category obtained from the PermCat-enriched Burnside category $G Burn_{pc}$ (Def. ) under replacing each hom-permutative category $Core(G FinSet_{sk}/_{S_1 \times S_2})$ with its Grothendieck group, hence with the abelian group which is the group completion

$$K \;\colon\; PermCat \overset{\text{iso classes}}{\longrightarrow} CMon \overset{K}{\longrightarrow} Ab$$

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

$$G Burn_{ab} \;\coloneqq\; K_{\bullet} G Burn_{pc} \,.$$

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 $A(G)$:

$$End_{G Burn_{ad}}(\ast, \ast) \;\simeq\; A(G)$$

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.)

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

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

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

This is called the spectral Burnside category.

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

Properties

Relation to $G$-Spectra

Proposition

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

There is a zig-zag of Quillen equivalences

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

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 $E$ assigns to the transitive G-set $G/H$ the fixed point spectrum $E^H$:

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

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

Related concepts

• Burnside ring

• Mackey functor, equivariant stable homotopy theory

• disjunctive (∞,1)-category

References

Lecture notes include:

• Andrew Blumberg, section 3.1 of The Burnside category, 2017 (pdf, GitHub)

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

• Bert Guillou, Peter May, Models of $G$-spectra as presheaves of spectra, (arXiv:1110.3571)

A beautified review is given in

• Anna Marie Bohmann, Angélica Osorno, section 7 of Constructing equivariant spectra via categorical Mackey functors, Algebraic & Geometric Topology 15.1 (2015): 537-563 (arXiv:1405.6126)

making use of

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