Schwede Global Equivariant K-theory

• Schwede Global Equivariant K-theory is a framework that generalizes algebraic K-theory to incorporate finite group actions, incomplete universes, and transfer systems.
• It employs equivariant Γ_G-spaces and Segal maps alongside bar constructions to assemble I-special data into connective equivariant spectra.
• The approach unifies classical nonequivariant K-theory and Shimakawa models by leveraging normed permutative G-categories and group-completion techniques.

Schwede Global Equivariant K-theory is a homotopical framework generalizing algebraic K-theory to capture equivariant structure relative to a finite group $G$, incomplete universes, and associated transfer systems. Central to this approach are Segal-type models for equivariant infinite loop spaces, generalizations of $\Gamma$-spaces, and normed permutative $G$-categories. This formulation unifies and extends the classical nonequivariant Segal $K$-theory construction and the Shimakawa model for genuine equivariant infinite loop spaces, with applications to equivariant $K$-theory of normed categories and the associated spectra (Bantelmann, 28 Oct 2025).

1. The Category $\Gamma_G$ of Finite Based $G$-Sets

The foundational indexing category in Schwede Global Equivariant K-theory is the category $\Gamma_G$ of finite based $G$-sets, as introduced by Shimakawa. Its structure is as follows:

Aspect	Description	Mathematical Representation
Objects	Finite based $G$-sets	$T_+ = T \sqcup \{*\}$, $T$: finite $G$-set
Morphisms	Based $G$-maps fixing the basepoint	$$\{\phi : T_+ \to S_+ \mid \phi(*) = *,\, \phi(g\cdot t) = g\cdot\phi(t)\}$$
Enrichment	Full $G\Top_*$-enriched subcategory	Objects: $T_+$; Morphisms: Based $G$-maps

The category's morphisms and composition laws are inherited from the category of based $G$-spaces, encoding simultaneous $G$-actions and basepoint structure. This categorical enrichment is essential for modeling equivariant infinite loop spaces indexed on incomplete universes.

2. Equivariant $\Gamma_G$-Spaces and the Segal Condition

An equivariant $\Gamma_G$-space is a $G\Top_*$-enriched functor $X: \Gamma_G \to G\Top_*$, which assigns to each $T_+$ a based $G$-space $X(T_+)$, and to each based $G$-map $\phi$ an equivariant map $X(\phi)$. For every injection $\phi: T_+\hookrightarrow S_+$, the map $X(\phi)$ is required to be a $(G \times \Sigma_\phi)$-cofibration, ensuring cofibrancy of the bar constructions required in later stages.

Segal maps are pivotal to encoding the "multiplicativity" of the structure. Given disjoint finite $G$-sets $S$ and $T$, the fold map induces

$$\varphi_{S,T}: X\bigl((S \sqcup T)_+\bigr) \longrightarrow X(S_+)\wedge X(T_+),$$

mirroring classical Segal's theory. The classical Segal maps correspond to families of coordinate projections,

$$\delta_n: X(n_+) \longrightarrow X(1_+)^n, \quad \delta_n(x) = (X(\delta_1)x,\, \dots,\, X(\delta_n)x),$$

capturing the expected equivalence to iterated products for special objects.

A $\Gamma_G$-$G$-space $X$ is termed $I$-special if for every $T\in I$ (where $I$ is a $G$-indexing system), the Segal map

$\delta_T: X(T_+) \xrightarrow{\simeq} X(1_+)^T$

is a $G$-equivalence. This property characterizes the types of $G$-sets that must be "inverted" homotopically, as dictated by the transfer system $I$ (Bantelmann, 28 Oct 2025).

3. The Segal Machine and Incomplete $G$-Spectra

Given a $G$-universe $U$ compatible with a transfer system $I$, the Segal machine describes a homotopically robust method for producing genuine or incomplete equivariant spectra from the data of $I$-special equivariant $\Gamma_I$-spaces. The construction is as follows:

• The bar-construction prolongation $b_I$ sends a $\Gamma_I$-space $X$ to a functor assigning

$A \mapsto B\bigl(A^\bullet,\, \Gamma_I,\, X\bigr),$

with $A^\bullet = \operatorname{Map}_*(-, A)$.

• The restriction $R_U^I$ evaluates at the representation spheres $S^V$ in $U$ and assembles structure maps

$$B\bigl((S^V)^\bullet,\, \Gamma_I,\, X\bigr)\wedge S^W \longrightarrow B\bigl((S^{V\oplus W})^\bullet,\, \Gamma_I,\, X\bigr).$$

• The composite

$$S_I^{G,U}: \Gamma_I[G_*]^{I\textrm{-spc}} \xrightarrow{b_I} Fun(G\underline{\;}_*^I, G_*) \xrightarrow{R_U^I} (\mathrm{Orth}^{G,U})$$

yields a connective positive $\Omega$-$G$-spectrum indexed on $U$ whenever $X$ is $I$-special.

The induced map from $X(1_+)$ to the associated $0$th space,

$$X(1_+) \longrightarrow S_I^{G,U}X(0) = B(\Gamma_I, \Gamma_I, X)(1_+)\xrightarrow{\sim}\Omega^V S_I^{G,U}X(S^V),$$

is a group-completion for all $V \in U$ with $V^G \neq 0$. This establishes an equivalence between the homotopy categories of $I$-special $\Gamma_I$-spaces and connective $U$-spectra indexed on the compatible universe $U$ (Bantelmann, 28 Oct 2025).

4. $I$-Normed Permutative $G$-Categories and the Segal Machine

An $I$-normed permutative $G$-category consists of a $G_*$-internal permutative $G$-category $\mathcal{A}$, together with external $T$-norms

$$\oplus_T: \mathcal{A}^{\times T} \longrightarrow \mathcal{A}, \quad T\in I,$$

and coherent untwistors $v_T$ realizing equivalences $$\mathcal{A}^{\times T} \simeq \mathcal{A}^{\oplus T}$$. The construction of the $\Gamma_I$-$G$-category $\overline{\mathcal{A}}$ proceeds by:

• Building $\overline{\mathcal{A}}_n$ whose objects are systems $(A_s)$ indexed by subsets $s \subseteq \{1,\ldots,n\}$, with structure isomorphisms

$$a_{s,t}: A_s\oplus A_t\cong A_{s \cup t},\quad s\cap t = \emptyset,$$

with morphisms given by compatible families of arrows $\alpha_s: A_s\to A'_s$.

• Equivariant structure under $\Sigma_n$ is defined via permutation of the index set, and $G$-equivariance via the internal $G$-action on $\mathcal{A}$.
• Restricting to $\Gamma_I$, $$\overline{\mathcal{A}}(T_+) = (\overline{\mathcal{A}}_n)^\sigma$$ for $T \cong n^\sigma$.

The Segal functor

$$\delta: \overline{\mathcal{A}}(T_+)\longrightarrow (\mathcal{A}^{\times T})^\sigma$$

is an internal equivalence of categories whenever $T\in I$, confirming $\overline{\mathcal{A}}$ is $I$-special. Taking the nerve and realization produces an $I$-special $\Gamma_I$-$G$-space:

$B\overline{\mathcal{A}}: \Gamma_I\longrightarrow G\Top_*.$

The equivariant Segal $K$-theory spectrum is then defined as

$$K_I^{G,U}(\mathcal{A}):= S_I^{G,U}(B\overline{\mathcal{A}}),$$

recovering Shimakawa’s genuine or incomplete equivariant $K$-theory spectrum of the normed $G$-category $\mathcal{A}$ (Bantelmann, 28 Oct 2025).

5. Equivalence of Models and Homotopical Consequences

The construction outlined above yields comparisons and equivalences between different formulations of equivariant infinite loop spaces and $K$-theory:

• As $I$ varies between transfer systems, comparison theorems establish that for each $I$, the homotopy category of $I$-special $\Gamma_I$-spaces is equivalent to that of connective $U$-indexed $G$-spectra, where $U$ is the universe compatible with $I$.
• The group-completion property of the $0$th space map ensures that the homotopy-theoretic content of the $I$-special $\Gamma_I$-space is reflected, after passage to spectra, in the classic group-completion context.

This suggests that the Segal machine furnishes a universal mechanism for passing from permutative $G$-categories, filtered by transfer systems, to the spectrum-level geometry of equivariant $K$-theory, unifying classical and Shimakawa-style approaches. The formulation is both flexible—accommodating incomplete universes and normed structures—and fundamentally homotopical in nature (Bantelmann, 28 Oct 2025).

6. Relationship to Previous Frameworks and Research Directions

Schwede Global Equivariant K-theory as developed through the segmented $\Gamma_I$ model provides a self-contained account of Shimakawa’s equivariant infinite-loop and $K$-theory machine in a modern categorical language. The introduction of $I$-normed permutative $G$-categories and the associated bar constructions advances the program of modeling equivariant infinite loop spaces for incomplete universes. The framework highlights categorical enrichment under $G$-action and advances connections to transfer systems, equivariant spectra, and homotopy theory.

Ongoing research directions include refinement of transfer system classification, understanding the interplay with norm homotopy types, and extension to broader equivariant contexts—such as global equivariant stable homotopy theory—in line with the perspectives opened in (Bantelmann, 28 Oct 2025).