Multifunctorial Equivariant K-Theory

• Multifunctorial Equivariant K-theory is a refined approach that unifies algebraic and topological K-theories into a single, globally coherent spectrum with strict equivariance.
• It employs enriched categorical frameworks like symmetric monoidal G-categories and parsummable categories to encode additivity, multiplicativity, and external products in a multifunctorial context.
• The theory recovers classical invariants and isomorphisms, such as Bott periodicity and Thom isomorphism, while unifying constructions across algebraic, topological, and operator algebraic settings.

Multifunctorial equivariant $K$-theory is a highly structured refinement of algebraic and topological $K$-theory, designed to simultaneously encode genuine equivariant homotopy-theoretic invariants for inputs with actions of one or more symmetry groups. This theory replaces naive “$G$-actions” at the spectrum level by constructions that are strictly functorial and multivariable at the categorical, spectrum, and homotopic levels. It realizes a single global or multifunctorial object that packages the $K$-theories of all group actions, generalizes to actions by groupoids and higher symmetry types, encodes multiplicativity at the spectrum level, and unifies prominent constructions across algebraic and operator algebraic $K$-theory.

1. Foundational Principles and Definitions

The core innovation of multifunctorial equivariant $K$-theory lies in assembling the data of $K$-theory with group action into a genuine $G$-spectrum or more generally, a “global equivariant spectrum” equipped with multifunctorial or multicategorical structure. This is achieved by:

• Modeling input data (rings, categories, $C^*$-algebras) with strictly coherent group (or groupoid) actions, typically as symmetric monoidal $G$-categories or more refined structures (parsummable categories, graded bundles, etc.).
• Constructing a spectrum-valued functor which is strictly functorial/multifunctorial in all variables, such that the output for each group $G$ is a genuine $G$-spectrum, not merely a spectrum with a $G$-action.
• Ensuring that morphisms between inputs (including pseudo-equivariant functors, correspondences, or group homomorphisms) induce strictly equivariant maps of spectra, compatible with all multiplicities, transfers, restrictions, and external products.
• Encoding additivity, multiplicativity, and external product structures at the level of multicategories, allowing multifunctoriality in multiple inputs and coherence of products.

Several formulations have been given, including spectrum-valued functors on finite $G$-rings (Merling, 2015), parsummable categories yielding global spectra (Schwede, 2019), operator algebraic correspondences for groupoids (Moutuou, 2013), multicategory-based infinite loop space machines (Guillou et al., 2021), and symmetric monoidal diagram categories (Johnson et al., 2022).

2. Categorical Input Structures and Multifunctoriality

Different branches of the field use distinct categorical frameworks to organize the multifunctoriality and genuine equivariance of $K$-theory:

• Symmetric monoidal $G$-categories: For rings or exact categories with $G$-action, the twisted module category is promoted to a strict genuine permutative $G$-category through constructions such as the “functor of points” $(G,-)$ (Merling, 2015, Guillou et al., 2021).
• Parsummable and $I$-categories: In the global setting, a parsummable category is a tame $I$-category with an associative, commutative, unital, and $I$-equivariant sum functor. These serve as the universal input for global $K$-theory (Schwede, 2019).
• Permutative categories and multicategories: Multifunctionality is realized by equipping the input categories with multifunctorial, symmetric monoidal, or multicategory structures—crucial for encoding the multilinearity and coherence needed for multifunctorial infinite loop space machines (Guillou et al., 2021, Johnson et al., 2022).
• Correspondence categories for groupoids: For operator-algebraic settings, objects are $C^*$-algebras or bundles with generalized actions, and morphisms are equivariant correspondences (Hilbert bimodules) compatible with the symmetry (Moutuou, 2013).

All these categorical enhancements ensure that $K$-theory becomes a genuine multifunctor from the input category (or multicategory) to the category (or multicategory) of genuine $G$-spectra or global spectra, with structured external and internal products, transfers, and restrictions.

3. Construction of Multifunctional Equivariant $K$-Theory

The multifunctorial $K$-theory spectrum is produced by processing the input categorical data through a sequence of operadic, categorical, and infinite loop space machines, which can be summarized as follows:

• Operadic infinite loop space machines: For a chaotic $E_\infty^G$-operad $\mathscr O$ in $G$-categories, a five-stage process (operad to operators, prolongation to $G$-set categories, strictification, geometric realization, Segal machine) produces a multifunctor $$K_G: \mathrm{Mult}(\mathscr O) \to \mathrm{Mult}(\mathrm{Sp}_G)$$, yielding orthogonal $G$-spectra which are associative ring/multiplicative $G$-spectra if the input is monoidal (Guillou et al., 2021).
• Segal–Shimakawa machines: Applying the category of finite based $G$-sets and functorial bar constructions yields a machine whose output is a genuine $G$-spectrum and which is strictly functorial in the multicategory of input $G$-categories (Merling, 2015).
• Global algebraic $K$-theory spectra: For a parsummable category $C$, the symmetric spectrum $K^{gl}(C)$ is defined by evaluating on all finite sets, functorially in $C$, and structured so that for each finite $G$ one obtains $K$-theory with genuine $G$-equivariance (Schwede, 2019).
• KKR bifunctors and operator-algebraic settings: In the classification of twisted and groupoid equivariant $K$-theories, $KKR^\mathfrak{G}(A,B)$ is bifunctorial and multifunctorial in groupoids $\mathfrak{G}_1,\ldots,\mathfrak{G}_n$, $C^*$-algebras, and external products, retaining coherence for all classical and twisted variants (Moutuou, 2013).
• Diagram categories and model-equivalence: The Elmendorf–Mandell machine demonstrates strict multifunctorial equivalence of $K$-theory from permutative categories through $\mathcal{G}_*$-categories and simplicial sets to connective spectra (Johnson et al., 2022).

In all cases, strict coherence and naturality of all mapping and product structures are enforced at each stage, and the transfer, restriction, and external product structures are built into the multifunctorial target.

4. Additivity, Multiplicativity, and Multicategorical Structures

Multifunctorial equivariant $K$-theory is fundamentally distinguished by its treatment of multilinearity, coherence, and multiplicativity:

• Additivity: The box product $\boxtimes$ on parsummable categories induces direct-sum decompositions at the spectrum level, mirroring additivity properties familiar from non-equivariant $K$-theory (Schwede, 2019).
• External products and multilinearity: The Kasparov external (cup) product, Day convolution on permutative categories, and operator-multicategory mechanisms ensure that the spectrum functor respects multilinear operations and external products in all input slots (Moutuou, 2013, Guillou et al., 2021, Johnson et al., 2022). The necessary symmetric and multicategorical coherence diagrams commute strictly.
• Multiplicative refinement: The Barratt–Priddy–Quillen theorem can be promoted to the multiplicative/operadic context, giving a lax monoidal natural equivalence between the suspension spectrum and the output of the $K$-theory machine applied to free $\mathscr O$-algebras (Guillou et al., 2021).
• Functoriality and transfers: Genuine equivariance (in the sense of the global stable homotopy category), means that restriction, induction, and transfer along all group homomorphisms are built in, and functorial maps between input categories induce strictly compatible maps at the spectrum level (Schwede, 2019).

5. Recovery of Classical and Representation-Theoretic Invariants

Multifunctorial equivariant $K$-theory recovers and unifies a range of prominent constructions in both algebraic and topological settings:

• Equivariant topological $K$-theory and Real $K$-theory: For rings like $\mathbb{R}$ or $\mathbb{C}$ with trivial or conjugation action, the multifunctorial theory produces connective covers of the classical $ku_G$, $ko_G$, and Real $KR$-theory spectra (Merling, 2015).
• Fixed-point, homotopy fixed-point, and assembly maps: For Galois extensions and representation categories, the fixed points and homotopy fixed points of $K_G(R)$ recover classical $K$-theory of the base field and the Quillen–Lichtenbaum and representational assembly maps (Merling, 2015).
• Swan and representation $K$-theory: The global functor of Swan $K$-groups associates to each group $G$ the group completion of isomorphism classes of $G$-objects in the input category, recovering pre-global and global functor structures on equivariant algebraic $K$-theory (Schwede, 2019).
• Operator algebraic cases: For real groupoids and twisted $K$-theory, the universal formalism produces standard isomorphisms such as the Bott periodicity and Thom isomorphism in full generality within the multifunctorial machine (Moutuou, 2013).

The diagrammatic and multifunctorial perspectives assert that all such specializations arise from a single universal construction, with explicit identification maps at each stage.

6. Equivalences of Homotopy Theories and Global Structure

A foundational result is that the $K$-theory multifunctor, for appropriate input categories (e.g., permutative categories, symmetric monoidal $G$-categories), induces equivalences of homotopy theories at each stage: from input categories, via multicategorical diagram categories, to symmetric spectra. Homotopy inverse functors exist at every level, and all are compatible with multifunctoriality, multiplicativity, and the full global structure (Johnson et al., 2022).

This unified viewpoint ensures that all classical models of equivariant and global $K$-theory, and their multifunctorial and multicategorical enhancements, ultimately participate in a single equivalence class of homotopy theories, implementing the multifunctorial philosophy.

7. Implications and Unification

The development of multifunctorial equivariant $K$-theory consolidates disparate threads in algebraic, topological, and operator algebraic $K$-theory, and facilitates the use of highly structured multiplicative and symmetric monoidal machinery in equivariant and global stable homotopy theory. All transfer, restriction, product, and additivity maps, along with classical isomorphisms (such as the Bott and Thom isomorphisms), are integrated into a single formalism whose input can range from strict permutative categories to generalized groupoid correspondences. The result is a framework in which equivariant, global, and twisted $K$-theories are all realized as facets of a highly multifunctorial spectrum-valued functor, equipped with the full array of coherence and multilinearity required by modern homotopy theory (Merling, 2015, Schwede, 2019, Moutuou, 2013, Guillou et al., 2021, Johnson et al., 2022).