nLab Tambara functor

Idea

Tambara functors are algebraic structures similar to Mackey functors, but with multiplicative norm maps as well as additive transfer maps, and a rule governing their interaction. They were introduced by Tambara, as TNR-functors, to encode the relationship between Trace (additive transfer), Norm (multiplicative transfer) and Restriction maps in the representation theory and cohomology theory of finite groups (Tam93).

In the $G$-equivariant context for a finite group $G$, the role of abelian groups in non-equivariant algebra is played by Mackey functors. The category of Mackey functors is a closed symmetric monoidal category with symmetric monoidal product, the box product. In addition to the expected generalization of commutative rings to simply commutative monoids for the box product, there is a poset of generalizations of the notion of commutative rings to the $G$-equivariant context: the incomplete Tambara functors. These interpolate between Green functors, the ordinary commutative monoids for the box product, and Tambara functors. The distinguishing feature for [incomplete] Tambara functors is the presence of certain multiplicative transfer maps, called norm maps. For a Green functor, we have no norm maps; for a Tambara functor, we have norm maps for any pair of subgroups $H \subset K$ of $G$. (Hill 17)

A Mackey functor is represented by a “$G$-equivariant Eilenberg–MacLane spectrum”, … a Tambara functor is represented by a commutative “$G$-equivariant Eilenberg-MacLane ring spectrum” (AB15)

Definition

In parts

Given a sequence of maps of finite $G$-sets $X \xrightarrow{g} Y \xrightarrow{h} Z$, we define distributor diagram

$$\Delta(g,h) = \left( X \xleftarrow{p} A \xrightarrow{q} B \xrightarrow{r} Z \right)$$

by setting

$$B \coloneqq \left\{ (z,s) \mid z \in Z, s\colon h^{-1}(z) \rightarrow X, \text{ s.t. } gs = \mathrm{id}_{h^{-1}(z)} \right\}$$

$$A \coloneqq Y \times_Z B$$

and setting $p(y,s) = s(y)$, $q(y,s) = (h(y),s)$, and $r(z,s) = z$.

Definition

A semi-Tambara functor for the group $G$ is the data of

• a $G$-coefficient system of sets $R$ and

• two $G$-semi-Mackey functors $(R,T)$ and $(R,N)$ with coefficient system $R$,

subject to the distributivity condition that, for all distributors $\Delta(f,g) = (X \xleftarrow{p} A \xrightarrow{q} A \xrightarrow{r} Z)$, we have

$$N_g T_f = T_r N_q R_p.$$

A semi-Tambara functor is a Tambara functor if $(R,T)$ is a Mackey functor.

We often refer to the maps $R_f$ as restrictions, $T_f$ as transfers, and $N_f$ as norms.

Categorically

We define a 2-category to have

• objects: finite $G$ sets

• morphisms: bispan diagrams $I \xleftarrow{p} X \xrightarrow{f} Y \xrightarrow{q} J$ in finite $G$-sets

• 2-cells: isomorphisms of bispan diagrams

The identity morphisms and 2-cells are as one would expect, and composition is defined by the outer bispan diagram

whose blue inner diagram is defined by the distributor

$$\Delta(g,f) = \left( X \times_J Y \xleftarrow{p} Y \times_Z B \xrightarrow{q} B \xrightarrow{r} Z \right).$$

Definition

Let $\mathcal{C}$ by an ∞-category admitting finite products. Then, a $G$-semi-Tambara functor valued in $\mathcal{C}$ is a product preserving functor $\mathrm{Bispan}(\mathbb{F}_G) \rightarrow \mathcal{C}$. A $G$-semi-Tambara functor is a Tambara functor if its underlying additive semi-Mackey functor is a Mackey functor.

In this formalism, given a semi-Tambara functor $X$, restriction $R_K^H\colon X(H) \rightarrow X(K)$ is implemented by functoriality under the bispan

$$G/H \leftarrow G/K = G/K = G/K,$$

norms by

$$G/K = G/K \rightarrow G/H = G/H,$$

and transfers by

$$G/K = G/K = G/K \rightarrow G/H.$$

Examples

(under construction…)

Burnside rings, representation rings, zeroth stable homotopy group of a genuine equivariant $E_{\infty}$-ring.

the homotopy category of Eilenberg MacLane commutative ring spectra is equivalent to the category of Tambara functors. (Ull13)

Some other examples are related to Witt-Burnside functors, Witt rings in the sense of Dress and Siebeneicher.

Additive completion of semi-Tambara functors

The following is Proposition 13.23 of Strickland 12.

Proposition

The additive completion of semi-Mackey functors underlies a left adjoint $\mathrm{sTamb}(\mathcal{C}) \rightarrow \mathrm{Tamb}(\mathcal{C})$ to the forgetful functor.

The Borel construction

Given $S$ a semiring with $G$-action through homomorphisms, we have a coefficient system $S^{\bullet}$ whose $[G/H]$-value is the invariants $S^H$. We give this the additional structure of a semi-mackey functor by first using the semiring isomorphism

$$S^H \simeq \mathrm{Hom}^G(G/H,S)$$

under the pointwise semiring structure, then given a map of transitive $G$-sets $r\colon G/K \rightarrow G/H$, writing

$$T_K^H(f)(x) = \sum_{r(y) = x} f(y).$$

We then give this the structure of a Tambara functor along $q\colon G/K \rightarrow G/H$ by the formula

$$N_K^H(f)(x) = \prod_{q(y) = x} f(y)$$

In particular, we may view $T_K^H$ as being defined by a “left Kan extension” formula and $N_K^H$ via “right Kan extension.”

Note that restricting to the value on $G/e$ yields a functor $U\colon \mathrm{sTamb}_{G}(\mathcal{C}) \rightarrow \mathrm{sRing}_{G}(\mathcal{C})$. The following is Example 6.3 of [Strickland 12]

Proposition

The Borel construction is right adjoint to $U\colon \mathrm{sTamb}_{G}(\mathcal{C}) \rightarrow \mathrm{sRing}_{G}(\mathcal{C})$.

Representation Tambara functors

Given $H \subset G$ a subgroup, note that (isomorphism classes of) $H$-representations correspond with $G$-equivariant vector bundles over $[G/H]$:

$$\mathrm{Vect}_k^{G}([G/H]) \simeq \mathrm{Rep}^H.$$

Restriction gives these the structure of a coefficient system; we may give these a semiring structure under $(\oplus,\otimes)$. Moreover, we lift this to a semi-Tambara structure with transfers given by induction

$$T_K^H(V)_x \simeq \bigoplus_{r(y) = x} V_y$$

and norms given by tensor-induction

$$N_K^H(V)_x \simeq \bigotimes_{q(y) = x} V_y.$$

The Representation Tambara functor is the additive completion of this.

Burnside Tambara functors

Let $\mathcal{A}(S)$ be the Grothendieck group

$$\mathcal{A}(S) \coloneqq K_0 \left((\mathbb{F}_G)_{/S} \right).$$

This becomes a coefficient system under precomposition. Moreover, colimits in $\mathbb{F}_G$ are universal, so precomposition along a map of finite $G$-sets $S \rightarrow T$ possesses a both a left and right adjoint; the left adjoint to $[G/K] \rightarrow [G/H]$ is called induction and the right adjoint is called coinduction. The Burnside Tambara functor has coefficient system $\mathcal{A}(-)$, transfers given by induction, and norms given by coinduction.

An explicit example

Let $G = C_2 = \{ e, \sigma \}$. Then, a C_2-Mackey functor consists of the data

• an Abelian group $X$ with $C_2$-action,
• an Abelian group $Y$,
• a $C_2$-equivariant restriction homomorphism $R\colon Y \rightarrow X$ (under trivial action on $Y$), and
• a $C_2$-equivariant transfer homomorphism $T\colon X \rightarrow Y$,

subject to the double coset formula

$$RT(a) = a + \sigma a.$$

Thus, a $C_2$-Tambara functor consists of * a semiring $X$ with $C_2$-action, * a semiring $Y$, * a $C_2$-equivariant semiring homomorphism $R\colon Y \rightarrow X$ (under trivial action on $Y$), * a $C_2$-equivariant additive map $T\colon X \rightarrow Y$, and * a $C_2$-equivariant multiplicative map $N\colon X \rightarrow Y$,

subject to the double coset formulas

• $RT(a) = a + \sigma a$

• $RN(a) = a \cdot \sigma a$

and the distributivity formula

• $T(a R(b)) = T(a) b.$

We may explicitly define a $C_2$-Tambara funcgtor by underlying coefficient system semiring $X = \mathbb{Z}[\alpha] / \alpha^2$, $Y = \mathbb{Z}[\beta, \gamma] / (\beta^2, \beta \gamma, \gamma^2, 2\gamma)$, under trivial $C_2$-action and restriction map

$$R(i + j\beta + k\gamma) = i + 2j\alpha.$$

We may give this a Tambara structure by transfer

$$T(i + j\alpha) = 2i + j\beta$$

and norm

$$N(i + j\alpha) = i^2 + ij\beta + j^2 \gamma.$$

Related concepts

• Mackey functor
• polynomial functor
• Witt vectors
• N-∞ operad

References

Originally,

• Daisuke Tambara, On multiplicative transfer, Comm. Algebra 21 (1993), no. 4, 1393–1420 (pdf).

Other references in homotopy theory,

• Neil Strickland, Tambara Functors, arXiv:1205.2516

• Michael Hill, Derived Equivariant Algebraic Geometry, (lecture and notes)

• Kristen Mazur, On the Structure of Mackey Functors and Tambara Functors, (thesis)

• John Ullman, Tambara Functors and Commutative Ring Spectra, (arXiv:1304.4912)

• Michael Hill, On the Andre-Quillen homology of Tambara functors, (arXiv:1701.06219)

On variations of Tambara functors,

• Vigleik Angeltveit and Anna Marie Bohmann, Graded Tambara Functors, (arXiv:1504.00668)

• Michael Hill, Andrew Blumberg, Incomplete Tambara functors, (arXiv:1603.03292)

• Michael Hill, Andrew Blumberg, The right adjoint to the equivariant operadic forgetful functor on incomplete Tambara functors, (arXiv:1711.11246)

• Michael Hill, Andrew Blumberg, Bi-incomplete Tambara functors, (arXiv:2104.10521)

• Michael Hill, David Mehrle, J.D. Quigley: Free incomplete Tambara functors are almost never flat, International Mathematics Research Notices 2023 5 (2023) [arXiv:2105.11513, doi:10.1093/imrn/rnab361]

• David Chan?, Bi-incomplete Tambara functors as O-commutative monoids, (arXiv:2208.05555)

• Ben Spitz?, Norms of Generalized Mackey and Tambara Functors, (arXiv:2409.13131)

Relationship with Witt vectors,

• Morten Brun, Witt vectors and Tambara functors, (arXiv:math/0304495),

• Hiroyuki Nakaoka?, Tambarization of a Mackey functor and its application to the Witt-Burnside construction, (arXiv:1010.0812)

In algebra,

• Hiroyuki Nakaoka?, A generalization of The Dress construction for a Tambara functor, and polynomial Tambara functors, (arXiv:1012.1911)

• Hiroyuki Nakaoka?, Ideals of Tambara functors (arXiv:1101.5982),

• Hiroyuki Nakaoka?, On the fractions of semi-Mackey and Tambara functors (arXiv:1103.3991),

• Hiroyuki Nakaoka?, Biset transformations of Tambara functors (arXiv:1105.0714),

• Hiroyuki Nakaoka?, Spectrum of the Burnside Tambara functor on a cyclic $p$-group (arXiv:1301.1453).

• Maxine Calle, Sam Ginnett, The Spectrum of the Burnside Tambara Functor of a Cyclic Group, (arXiv:2011.04729)

• Noah Wisdom, A classification of $C_{p^n}$-Tambara fields, (arXiv:2409.02966)

• David Chan?, David Mehrle, J.D. Quigley, Ben Spitz?, Danika Van Niel?, On the Tambara Affine Line (arXiv:2410.23052)

In derived algebra,

• David Mehrle, J.D. Quigley, Michael Stahlhauer?: Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic $p$-Groups [arXiv:2407.18382]

• David Mehrle, J.D. Quigley, Michael Stahlhauer?, Pathological Computations of Mackey Functor-valued $Tor$ over Cyclic Groups [arXiv:2410.11974]

In higher algebra,

• Elden Elmanto?, Rune Haugseng, On distributivity in higher algebra I: The universal property of bispans, (arXiv:2010.15722)

• Bastiaan Cnossen, Rune Haugseng, Tobias Lenz, Sil Linskens, Normed equivariant ring spectra and higher Tambara functors, (arXiv:2407.08399)