Incomplete Tambara Functors

• Incomplete Tambara Functors are algebraic structures that extend Green functors by incorporating only those multiplicative norms specified by a chosen indexing system.
• They interpolate between Green functors without extra norms and complete Tambara functors with all norms, reflecting features of N∞ ring spectra.
• Their unique behavior is highlighted by free incomplete Tambara functors often being non-free and non-flat as Mackey functors, complicating module theory.

Incomplete Tambara functors are the algebraic structures that arise when a Green functor is endowed only with those multiplicative norms prescribed by a chosen indexing system. For a finite group $G$, they interpolate between Green functors, where no nontrivial norms are present, and complete Tambara functors, where all norms are present. They were introduced as the algebraic analogue of $N_\infty$ ring structures, motivated in part by the fact that Bousfield localization can destroy some of the norm maps of a genuine equivariant commutative ring spectrum, so that $\pi_0$ need not carry full Tambara structure (Blumberg et al., 2016). Their later study showed that free incomplete Tambara functors behave very differently from classical free algebras: free incomplete Tambara functors are often not free, and usually not even flat, as underlying Mackey functors (Hill et al., 2021).

1. Indexing systems and polynomial categories

Let $G$ be a finite group. An indexing system $O$ is a full symmetric monoidal sub-coefficient system of finite $H$-sets that contains all trivial sets, is closed under finite limits, and is closed under self-induction. Equivalently, one may work with an indexing category $O$ for $G$, namely a wide, pullback-stable, finite coproduct complete subcategory of the category of finite $G$-sets. The assignment $O \mapsto \mathrm{Set}^G_O$ gives an isomorphism of posets between indexing systems and wide, pullback-stable, finite coproduct-complete subcategories of the category of finite $N_\infty$0-sets (Blumberg et al., 2016).

Given such an indexing category, one forms the category of polynomials with exponents in $N_\infty$1, denoted $N_\infty$2. Its objects are finite $N_\infty$3-sets, and its morphisms are isomorphism classes of diagrams

$N_\infty$4

with $N_\infty$5. Every morphism admits the canonical decomposition

$N_\infty$6

Here $N_\infty$7, $N_\infty$8, and $N_\infty$9 are the restriction, transfer, and norm generators associated to a map $\pi_0$0. Pullbacks and exponential diagrams govern composition, and the canonical commutation relations remain internal to $\pi_0$1 because $\pi_0$2 is pullback-stable (Blumberg et al., 2016).

The two extremal indexing categories are fundamental. The trivial indexing category $\pi_0$3 contains only isotropy-preserving maps, and $\pi_0$4-Tambara functors are Green functors. The complete indexing category $\pi_0$5 yields complete Tambara functors, with all norms present (Hill et al., 2021).

2. Structure maps and reciprocity

An $\pi_0$6-Tambara functor is a product-preserving functor

$\pi_0$7

such that each $\pi_0$8 is an abelian group, in fact a commutative ring. For an orbit $\pi_0$9, $G$0 is a commutative $G$1-ring, where $G$2 acts by conjugation. If $G$3 and $G$4 is the orbit map, then the structure maps are

$G$5

with the norm $G$6 present precisely when $G$7 is admissible in $G$8 (Hill et al., 2021).

The basic identities are the pullback formulas

$G$9

together with the exponential-diagram identity expressing distributivity of norms over transfers. On values, these identities encode Mackey double-coset compatibility, Frobenius reciprocity for transfers,

$O$0

and Tambara reciprocity for norms (Blumberg et al., 2016).

For admissible $O$1, the classical formula for a norm on rings is

$O$2

The incomplete setting retains exactly those norm maps and reciprocity relations that are licensed by the indexing system. This is the source of the term “incomplete”: the additive Mackey structure is still present, but multiplicative transfers occur only along admissible exponents (Blumberg et al., 2016).

3. Free objects, forgetful functors, and module theory

The Burnside category $O$3 is the additive completion of the category of spans of finite $O$4-sets, and a $O$5-Mackey functor is an additive functor $O$6. Equivalently, a Mackey functor is a product-preserving functor $O$7 landing in abelian groups. Restriction along the inclusion $O$8 gives the underlying Mackey functor

$O$9

Thus every incomplete Tambara functor has an underlying Mackey functor, and the passage from Tambara data to Mackey data is functorial (Hill et al., 2021).

For a finite $H$0-set $H$1, the free $H$2-Tambara functor on one generator at level $H$3 is

$H$4

the group completion of the represented functor. It satisfies the universal property

$H$5

If $H$6 is any $H$7-Tambara functor, the free $H$8-algebra on one generator at level $H$9 is

$O$0

where $O$1 is the box product of Mackey functors (Hill et al., 2021).

The box product makes Mackey functors into a symmetric monoidal category with unit $O$2, the Burnside Mackey functor. If $O$3 is a Green functor, then an $O$4-module is a Mackey functor $O$5 equipped with an action map $O$6. In $O$7, “free” means a direct sum of $O$8’s, “projective” means a summand of a free, and “flat” means exactness of $O$9. Base change along a map $G$0 preserves free, projective, and flat objects (Hill et al., 2021).

The category of incomplete Tambara functors is complete and cocomplete. The forgetful functor from $G$1-Tambara functors to Mackey functors is strong symmetric monoidal and has a left adjoint $G$2, given by left Kan extension. Change-of-indexing-system and change-of-groups functors admit adjoints as well: if $G$3, there is a forgetful functor $G$4 with left adjoint given by Kan extension, and for $G$5, restriction $G$6 has both a right adjoint $G$7 and a left adjoint $G$8 (Blumberg et al., 2016). For finite cyclic groups of prime order, the right adjoint to the operadic forgetful functor admits an explicit description in terms of pullbacks (Blumberg et al., 2017).

4. Freeness, flatness, and the failure of the classical analogy

Classically, free algebras are always free as modules over the base ring. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. A central result is that free incomplete Tambara functors often fail to be free as Mackey functors, and for solvable groups the precise flatness criterion is very restrictive (Hill et al., 2021).

A sufficient criterion for freeness is available. If $G$9, $G$0, and $G$1 is admissible in $G$2, then the free $G$3-Tambara functor $G$4 is free as an $G$5-module, hence free as a Mackey functor. The proof uses the norm left adjoint $G$6 and the fact that norms preserve freeness in the relevant setting (Hill et al., 2021).

The converse direction is sharper. If $G$7 is flat as an $G$8-module, then $G$9 and $O \mapsto \mathrm{Set}^G_O$0 is normal in $O \mapsto \mathrm{Set}^G_O$1. Moreover, if $O \mapsto \mathrm{Set}^G_O$2 is solvable, then $O \mapsto \mathrm{Set}^G_O$3 is admissible in $O \mapsto \mathrm{Set}^G_O$4. For solvable $O \mapsto \mathrm{Set}^G_O$5, the following are equivalent:

1. the underlying Mackey functor $O \mapsto \mathrm{Set}^G_O$6 is flat;
2. it is free;
3. $O \mapsto \mathrm{Set}^G_O$7, $O \mapsto \mathrm{Set}^G_O$8 is admissible in $O \mapsto \mathrm{Set}^G_O$9, and $N_\infty$00 (Hill et al., 2021).

The obstruction comes from norms. The key mechanism is that norms impose nontrivial multiplicative relations tied to admissible sets, and symmetric powers appearing in Green functors lead to non-flat summands. In particular, if $N_\infty$01, then $N_\infty$02 is not flat. The proof reduces, after restriction and passage to cyclic $N_\infty$03-groups, to the fact that for $N_\infty$04, neither the constant Mackey functor nor the dual constant Mackey functor is flat (Hill et al., 2021).

This failure is asymptotically dominant. If $N_\infty$05 counts pairs $N_\infty$06 and $N_\infty$07 counts those pairs yielding flat underlying Mackey functors, then

$N_\infty$08

For solvable $N_\infty$09, the bound $N_\infty$10, where $N_\infty$11 is the depth of the subgroup lattice, implies that free incomplete Tambara functors are almost never flat as Mackey functors (Hill et al., 2021).

5. Localization and representative examples

The global failure of flatness admits a precise localization remedy. Let $N_\infty$12 denote the localization of the Burnside Tambara functor at the class $N_\infty$13. Then there is an isomorphism of Tambara functors

$N_\infty$14

and any $N_\infty$15-module $N_\infty$16 is isomorphic to the fixed point functor of its underlying module $N_\infty$17. Equivalently,

$N_\infty$18

In particular, for any indexing category $N_\infty$19 and subgroup $N_\infty$20, the localized free object $N_\infty$21 is free as an $N_\infty$22-module (Hill et al., 2021).

For $N_\infty$23, the Burnside Tambara functor has

$N_\infty$24

with $N_\infty$25, transfer $N_\infty$26 given by multiplication by $N_\infty$27, and norm

$N_\infty$28

This formula makes visible the torsion-type terms introduced by norms (Hill et al., 2021).

Small-group examples show the sharpness of the freeness criterion. For $N_\infty$29, the indexing categories are $N_\infty$30 and $N_\infty$31; $N_\infty$32 is free, while $N_\infty$33 is not free. For $N_\infty$34, the pattern remains sparse: freeness persists only for generators at normal subgroups with trivial restriction of $N_\infty$35 and admissibility as dictated by the theorem. For the dihedral group $N_\infty$36, freeness of $N_\infty$37 occurs only where $N_\infty$38 is normal, $N_\infty$39 restricts trivially at $N_\infty$40, and $N_\infty$41 is admissible; after localization at $N_\infty$42, all such examples become free (Hill et al., 2021).

6. Relation to $N_\infty$43 ring spectra and later developments

Incomplete Tambara functors arise as $N_\infty$44 of $N_\infty$45 algebras. If $N_\infty$46 is an $N_\infty$47 operad and $N_\infty$48 is an $N_\infty$49-algebra in orthogonal $N_\infty$50-spectra, then $N_\infty$51 is an $N_\infty$52-Tambara functor. The allowed norms on $N_\infty$53 are exactly those prescribed by $N_\infty$54. This gives the algebraic reflection of $N_\infty$55 operads at the level of $N_\infty$56, and it explains why Bousfield localization can produce algebraic structures lying strictly between Green functors and complete Tambara functors (Blumberg et al., 2016).

A later extension restricts additive transfers as well as multiplicative norms. In the bi-incomplete setting, one works with a compatible pair $N_\infty$57 of transfer systems or indexing categories, where $N_\infty$58 controls norms and $N_\infty$59 controls transfers. Coefficient systems, Green functors, and full Tambara functors then become the cases $N_\infty$60, $N_\infty$61, and $N_\infty$62, and bi-incomplete Tambara functors are characterized as $N_\infty$63-commutative monoids in $N_\infty$64-Mackey functors (Blumberg et al., 2021, Chan, 2022). Prime ideals and spectra have also been extended to this setting, simultaneously generalizing Lewis’s notion for Green functors and Nakaoka’s notion for Tambara functors (Balchin et al., 8 May 2026).

Homological algebra over free incomplete Tambara functors has developed in parallel. For odd primes $N_\infty$65, cyclic-$N_\infty$66-group-equivariant analogues of the Koszul resolution resolve the Burnside Mackey functor as a module over free incomplete Tambara functors, and for free incomplete Tambara functors whose underlying Mackey functor is projective there is an isomorphism

$N_\infty$67

These constructions are used to compute Mackey functor-valued Hochschild homology for cyclic groups of odd prime order and for $N_\infty$68 (Mehrle et al., 2024). By contrast, later computations over cyclic groups exhibit pathological behavior: unlike the classical polynomial ring case, where $N_\infty$69 vanishes above degree one, there are examples in which Mackey functor-valued $N_\infty$70 is nonvanishing in almost every degree (Mehrle et al., 2024). This suggests that incomplete Tambara functors retain the formal role of equivariant polynomial algebras while exhibiting substantially different homological behavior from their nonequivariant analogues.