Spectral Mackey Functors

• Spectral Mackey functors are homotopy-coherent versions of classical Mackey functors that encode restriction, transfer, and conjugation within a unified stable object.
• They organize functorial data from effective Burnside and span categories, enabling precise ∞-categorical formulations for equivariant and motivic homotopy theory.
• Their structure supports excision, symmetric monoidal operations, and universal constructions that link algebraic K-theory, derived categories, and polynomial functor calculus.

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. In the modern $\infty$-categorical formulation, they are organized as direct-sum-preserving or product-preserving functors out of an effective Burnside or span category into spectra, and for finite groups they provide a model for genuine equivariant spectra. They therefore package restriction, transfer, conjugation, and the higher coherences relating these operations in a single stable object, rather than as separate algebraic axioms (Barwick, 2014, Lenz, 15 Aug 2025).

1. Definition through spans and effective Burnside categories

The basic input for the general theory is a disjunctive triple $(C, C_{\dag}, C^{\dag})$, consisting of an $\infty$-category $C$ together with subcategories of ingressive and egressive morphisms, both containing the equivalences. From this data one forms the effective Burnside $\infty$-category $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$. Its objects are the objects of $C$, while its morphisms are spans

$$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$

composed by pullback. This is the ambient $\infty$-category that records both covariant and contravariant functoriality and the corresponding base-change structure (Barwick, 2014).

A spectral Mackey functor with values in an additive $\infty$-category $(C, C_{\dag}, C^{\dag})$0, and in particular in spectra, is then a direct-sum-preserving functor

$(C, C_{\dag}, C^{\dag})$1

For a finite group $(C, C_{\dag}, C^{\dag})$2, if $(C, C_{\dag}, C^{\dag})$3 denotes the category of finite $(C, C_{\dag}, C^{\dag})$4-sets, the corresponding category of spectral Mackey functors can be written as

$(C, C_{\dag}, C^{\dag})$5

and this category is equivalent to the $(C, C_{\dag}, C^{\dag})$6-category of genuine $(C, C_{\dag}, C^{\dag})$7-spectra. In this form, equivariant spectra are recovered entirely inside the language of span categories and product-preserving functors (Lenz, 15 Aug 2025).

This formulation is not merely a repackaging of ordinary Mackey functors with values in spectra. A central point of the theory is that the domain is itself an $(C, C_{\dag}, C^{\dag})$8-categorical Burnside object, so the higher homotopies governing pullbacks, compositions, and double-coset phenomena are built into the definition. That feature is what makes spectral Mackey functors a genuine homotopy-coherent generalization rather than a pointwise enrichment of the classical notion (Barwick, 2014).

2. Excision, coherence, and universal constructions

A major structural result is that spectral Mackey functors can be described as excisive functors on a suitable $(C, C_{\dag}, C^{\dag})$9-category. More precisely, Barwick identifies them with $\infty$0-excisive functors from a derived $\infty$1-category associated to the effective Burnside category. This places spectral Mackey functors directly inside the framework of Goodwillie-style excision and explains why they behave as the linear objects attached to equivariant span data (Barwick, 2014).

The same work introduces the unfurling construction for families of Waldhausen $\infty$2-categories equipped with suitable adjoint pairs of functors. The point of unfurling is to convert bicartesian Waldhausen fibrations, in which the required compatibility data are only locally visible, into cocartesian fibrations over the effective Burnside $\infty$3-category. The result is a spectral Mackey functor valued in Waldhausen $\infty$4-categories and, after algebraic $\infty$5-theory, in spectra. In the language of the paper, this completely solves the homotopy coherence problem that arises when one wishes to study the algebraic $\infty$6-theory of such objects as spectral Mackey functors (Barwick, 2014).

These coherence results also explain why algebraic $\infty$7-theory occupies a universal role in the subject. Barwick uses the same framework to show that universal examples of spectral Mackey functors are given by algebraic $\infty$8-theory, and the resulting constructions support assembly and coassembly morphisms as well as Segal–tom Dieck type splitting phenomena in examples ranging from profinite groups to $\infty$9-theory and derived stacks (Barwick, 2014).

3. Symmetric monoidal structure, norms, and Tambara phenomena

The higher algebra of spectral Mackey functors becomes visible once the effective Burnside $C$0-category is endowed with symmetric promonoidal structure. Using symmetric promonoidal $C$1-categories and a generalization of Day convolution, Barwick, Glasman, and Shah construct a symmetric monoidal structure on the $C$2-category of Mackey functors. In particular, for functors $C$3 one has the convolution formula

$C$4

and this makes it possible to speak of spectral Green functors for any operad $C$5. The same paper proves that the algebraic $C$6-theory of group actions is lax symmetric monoidal and gives a short proof of the equivariant Barratt–Priddy–Quillen theorem, identifying the algebraic $C$7-theory of the category of finite $C$8-sets with the $C$9-equivariant sphere spectrum (Barwick et al., 2015).

At the level of $\infty$0, these multiplicative structures recover the algebraic norm and Tambara operations studied for Mackey functors. For a Mackey functor $\infty$1, Ullman describes the derived symmetric power by

$\infty$2

the derived norm by

$\infty$3

and the $\infty$4-symmetric monoidal power by

$\infty$5

Algebraically, $\infty$6 is identified with the degree $\infty$7 piece of the free Tambara functor, the norm is modeled by explicit generators and relations, and Tambara functors are characterized as Mackey functors equipped with multiplicative pushforwards satisfying coherence axioms (Ullman, 2013).

This relation between spectral and algebraic constructions is fundamental. The paper repeatedly emphasizes that the algebraic models provide explicit formulas for the zeroth homotopy Mackey functor of the corresponding spectral constructions. A common misconception is that Tambara structure is external to spectral Mackey functors; the algebraic description instead shows that norms, symmetric powers, and multiplicative pushforwards arise as $\infty$8-shadows of spectral operations (Ullman, 2013).

4. Categorical, 2-categorical, and enriched extensions

The spectral theory sits inside a broader hierarchy of categorical Mackey structures. Balmer and Dell’Ambrogio define Mackey 2-functors as strict 2-functors

$\infty$9

satisfying additivity, induction and coinduction, base-change for iso-comma squares, and ambidexterity. They then construct a universal 2-category of Mackey 2-motives from spans and spans of spans, show that every Mackey 2-functor factors through it, and identify the 2-endomorphism ring of the identity of $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$0 with the crossed Burnside ring of $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$1. The same work explicitly notes similarities and connections between this formalism and the universal properties for spectral Mackey functors (Balmer et al., 2018).

The survey on Mackey and Green 2-functors places this in a decategorification sequence. In a Green 2-functor each category $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$2 carries a symmetric monoidal structure, the restriction functors are strong monoidal, and induction satisfies a projection formula such as

$A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$3

Classical Mackey and Green functors are then recovered by passing to $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$4 or Hom-groups. This clarifies that the 2-categorical level preserves the adjunctions, natural transformations, and coherence data that are invisible after decategorification (Dell'Ambrogio, 2023).

A second route to spectral Mackey functors proceeds through enrichment. Bohmann and collaborators develop Mackey functors enriched over closed multicategories and study change of enrichment along $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$5-theory multifunctors. If $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$6 is such a multifunctor, then permutative-category-enriched Mackey functors can be sent to spectral Mackey functors by changing the enrichment to symmetric spectra. This is closely related to Guillou–May type models of genuine $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$7-spectra as spectral Mackey functors (Johnson et al., 2022).

The construction of equivariant spectra via categorical Mackey functors makes this explicit. Given a $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$8-functor $A^{\mathrm{eff}}(C, C_{\dag}, C^{\dag})$9, there is a functor

$C$0

producing a spectral Mackey functor and hence a genuine equivariant spectrum. As applications, the same machine yields functorial constructions of equivariant Eilenberg–MacLane spectra for Mackey functors and suspension spectra for finite $C$1-sets (Bohmann et al., 2014).

5. Motivic and algebraic $C$2-theoretic realizations

The motivic analogue of the theory identifies genuine equivariant motivic spectra with stabilized Mackey-style transfer objects. For a finite constant group $C$3 acting on a scheme $C$4 such that $C$5 is invertible in the residue fields of $C$6, Bachmann proves the equivalence

$C$7

where $C$8 is the category of motivic spaces with finite étale transfers over the quotient stack $C$9, and $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$0 is the trivial representation sphere. The paper constructs norm functors $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$1 for finite étale and suitable proper étale morphisms, proves formulas such as

$$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$2

and extends the homotopy $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$3-structure to DM-stacks in a way compatible with norms and transfers. In this setting, spectral Mackey functors are realized through spans with finite étale morphisms rather than through orbit categories of finite groups (Bachmann, 2022).

Equivariant algebraic $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$4-theory supplies a concrete computational class of examples. For a finite group $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$5 acting on a ring $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$6, the equivariant algebraic $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$7-theory spectrum can be interpreted as a spectral Mackey functor, and its $$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$8th homotopy Mackey functor is computed algebraically by

$$X \xleftarrow{\mathrm{egressive}} U \xrightarrow{\mathrm{ingressive}} Y$$9

where $\infty$0 is the twisted group ring. Restriction, transfer, and conjugation are given respectively by restriction of scalars, extension of scalars, and the isomorphisms induced by conjugation on twisted group rings, yielding an explicit algebraic model for the homotopy Mackey functors of Merling’s equivariant $\infty$1-theory spectrum (Brazelton, 2021).

A recent $\infty$2-categorical reformulation replaces point-set constructions by the monoidal Borel construction. For a finite group $\infty$3, one obtains a right adjoint

$\infty$4

and the paper shows that the spectral Mackey functors associated to the equivariant algebraic $\infty$5-theory spectra of Guillou–May and Merling are canonically described by applying non-equivariant algebraic $\infty$6-theory pointwise after this monoidal Borel construction. Pützstück’s global version gives the corresponding statement for global algebraic $\infty$7-theory spectra (Lenz, 15 Aug 2025).

6. Derived, chromatic, and polynomial perspectives

Derived Mackey functors form a closely related but strictly weaker invariant than genuine equivariant stable homotopy theory. Patchkoria, Sanders, and Wimmer compute the Balmer spectrum of the category of derived Mackey functors for all finite groups and show that, as a set,

$\infty$8

They further show that this spectrum captures precisely the top and bottom layers, namely the height $\infty$9 and height $\infty$0 parts, of the spectrum of the equivariant stable homotopy category. The same paper identifies Kaledin’s category with the derived category of modules over the equivariant ring spectrum $\infty$1, proves symmetric monoidal equivalences with Barwick’s spectral Mackey functors, and contrasts this with the ordinary derived category of Mackey functors, whose geometric fixed points behave differently (Patchkoria et al., 2020).

Tensor triangular geometry adds a classification theory for spectral Mackey functor categories. Barthel, Castellana, Heard, Sanders, and Valenzuela prove that if $\infty$2 is stratified and $\infty$3 is noetherian, then $\infty$4 is stratified for every finite group $\infty$5. For $\infty$6-local spectral Mackey functors they obtain

$\infty$7

as a set, and they completely classify the localizing tensor-ideals of these categories. This gives a full classification in settings where the corresponding problem for the full equivariant stable homotopy category remains open (Barthel et al., 2021).

A different generalization replaces orbit categories by surjection categories. Glasman proves that the category of $\infty$8-excisive functors from finite spectra to a stable $\infty$9-category $(C, C_{\dag}, C^{\dag})$00 is equivalent to the category of $(C, C_{\dag}, C^{\dag})$01-valued Mackey functors on the indexing category of finite sets of cardinality at most $(C, C_{\dag}, C^{\dag})$02 and surjections, with cross-effects furnishing the corresponding Mackey data (Glasman, 2016). A recent multiplicative strengthening states that reduced $(C, C_{\dag}, C^{\dag})$03-excisive endofunctors on spectra satisfy

$(C, C_{\dag}, C^{\dag})$04

as symmetric monoidal categories, with Day convolution on both sides, and extends this to multivariable and subdiagonal functors (Barthel et al., 2 Apr 2026).

These developments show that spectral Mackey functors are not confined to classical equivariant stable homotopy theory. They also organize motivic transfers, equivariant and global algebraic $(C, C_{\dag}, C^{\dag})$05-theory, derived Mackey categories, and polynomial functor calculus. A plausible implication is that the span-and-excision formalism is best understood as a general mechanism for encoding ambidextrous functoriality together with its multiplicative and stable refinements, rather than as a device tied only to finite group actions.