Bredon sheaf cohomology

Abstract: For a finite group $G$, we compute the algebraic $K$-theory of the category of equivariant sheaves on a locally compact Hausdorff $G$-space, generalizing a result of Efimov, and determine the equivariant $E$-theory of the $C^*$-algebra of continuous functions. These invariants admit natural descriptions in terms of a new equivariant cohomology theory, which we call Bredon sheaf cohomology. This theory recovers classical Bredon cohomology for $G$-CW complexes and ordinary sheaf cohomology when $G$ is trivial. We establish its basic structural properties and prove a strong uniqueness theorem: any functor from the category of locally compact Hausdorff $G$-spaces to a dualizable stable category satisfying equivariant open descent and cofiltered compact codescent is equivalent to Bredon sheaf cohomology, generalizing a result of Clausen.

• The paper introduces Bredon sheaf cohomology, unifying classical Bredon and sheaf cohomology into a universal equivariant theory for finite group actions.
• It establishes key structural properties such as open descent, cofiltered compact codescent, and G-homotopy invariance, ensuring computational viability.
• The work bridges algebraic and topological K-theory by demonstrating equivalences via computed spectra and a geometric exit-path approach.

Bredon Sheaf Cohomology: Synthesis and Uniqueness in Equivariant Topology

Introduction and Motivation

The paper "Bredon sheaf cohomology" (2604.08066) articulates a new synthesis between classical Bredon cohomology and sheaf cohomology, resulting in an equivariant sheaf cohomology theory for topological $G$-spaces where $G$ is a finite group. This theory, named Bredon sheaf cohomology, provides an explicit geometric and sheaf-theoretic refinement of Bredon cohomology, with substantial applications to equivariant algebraic $K$-theory, topological $K$-theory, and noncommutative motives. The authors establish not only foundational structural results but a strong uniqueness theorem that positions Bredon sheaf cohomology as universal among functors satisfying equivariant open descent and cofiltered compact codescent.

Formal Definition and Main Theorem

Let $G$ be a finite group and $X$ a locally compact Hausdorff $G$-space. For a dualizable stable $\infty$-category $\mathcal{C}$, a $G$-equivariant sheaf is an object of $G$0, the homotopy fixed point category under the $G$1-action. Given a coefficient system $G$2 (a functor from the $G$3-orbit category to spectra), define the Bredon sheaf cohomology of $G$4 with coefficients in $G$5 as

$G$6

where $G$7 is the sheafification (on the orbit space $G$8) of the presheaf taking $G$9 to the colimit over equivariant maps $K$0 (with $K$1 a $K$2-orbit) of $K$3.

Main equivalence: For locally compact Hausdorff $K$4-spaces $K$5,

$K$6

where $K$7 is a coefficient system assigning to $K$8 the $K$9-theory spectrum of $K$0 [(2604.08066), Thm. A]. This result is a Bredon-theoretic refinement of Efimov's computation in the non-equivariant case, extending the bridge between algebraic $K$1-theory and sheaf theory into the equivariant domain.

Structural Properties

Bredon sheaf cohomology enjoys the following properties:

1. Normalization: For every $K$2-orbit $K$3, $K$4.
2. Open Descent: The assignment is a sheaf for the Grothendieck topology generated by $K$5-invariant open covers.
3. Cofiltered Compact Codescent: For cofiltered limits of compact $K$6-spaces $K$7, the natural map $K$8 is an equivalence; essential technical input is provided by Abels' slice theorem.
4. $K$9-Homotopy Invariance: $G$0-homotopy equivalences induce equivalences on cohomology.
5. Compatibility with Singular Bredon Cohomology: For $G$1-CW complexes, their Bredon sheaf cohomology canonically agrees with the classical singular Bredon cohomology.

Uniqueness and Universality

The theory is uniquely characterized as follows: any functor $G$2 (with $G$3 a dualizable stable category) that satisfies open descent and cofiltered compact codescent is naturally equivalent to Bredon sheaf cohomology with appropriate coefficients. Explicitly,

$G$4

via $G$5. Thus, Bredon sheaf cohomology is universal among homotopy-invariant, descent-satisfying equivariant cohomology theories, outstripping the possibility of non-homotopy-invariant theories with these properties.

Applications: Equivariant Algebraic K-theory and E-theory

By lifting Efimov’s algebraic $G$6-theory correspondence to the equivariant setting, the authors obtain computational results for the $G$7-theory of $G$8-equivariant sheaf categories and $G$9-algebras of functions:

• Algebraic $X$0-theory: The $X$1-theory of $X$2 is given by the compactly supported Bredon sheaf cohomology of $X$3 with coefficients in $X$4.
• Topological $X$5-theory of Crossed Products: There is a natural equivalence (for $X$6 finite, $X$7 locally compact Hausdorff):

$X$8

where $X$9 is the coefficient system sending $G$0 to $G$1.

Furthermore, the formalism applies to noncommutative motives and equivariant $G$2-theory, via their respective universal localizing properties and descent features.

Geometric Description: Constructibility and Computability

For $G$3 a locally compact Hausdorff $G$4-space, the sheaf $G$5 is constructible with respect to the orbit-type stratification of $G$6. The stalk at the point corresponding to the orbit $G$7 is canonically $G$8. For $G$9-manifolds or $\infty$0-spaces with finitely many orbit types, Bredon sheaf cohomology is computable as a limit over the exit-path category of the stratified space $\infty$1, functorially determined by the orbit data and the coefficient system. This provides effective reduction of global calculations to local and incidence data, rendering explicit computation feasible in combinatorial and geometric examples.

Cohomology–Homotopy Comparison and the Equivariant Shape

The authors identify Bredon sheaf cohomology with singular Bredon cohomology for $\infty$2-CW complexes and more generally for spaces whose orbit space $\infty$3 is hypercomplete. A categorical shape theory is introduced: for every $\infty$4-space $\infty$5, its equivariant shape is a pro-$\infty$6-anima functor $\infty$7, such that

$\infty$8

where $\infty$9 is the singular Bredon cohomology and the colimit is over a suitable pro-system approximating $\mathcal{C}$0. This construction situates Bredon sheaf cohomology within the context of equivariant homotopy theory and topos-theoretic shape, integrating geometric and categorical perspectives.

Bold Results and Theoretical Implications

• Strong Uniqueness Theorem: No alternative equivariant sheaf cohomology theory with open descent and compact codescent can fail to be homotopy invariant or disagree with Bredon sheaf cohomology on orbit spaces.
• Parallelisms between Dualizable Categories and $\mathcal{C}$1-Algebras: The analogy extends through theorems connecting equivariant algebraic $\mathcal{C}$2-theory and topological $\mathcal{C}$3-theory of crossed products to Bredon sheaf cohomology, underlying a unification in the formalism of algebraic and topological invariants of $\mathcal{C}$4-spaces.
• Reduction to Exit-path Computation: The geometric incarnation of Bredon sheaf cohomology as limits over exit-path categories clarifies the computational structure for stratified orbit spaces.

Future Directions

This theory opens further research routes in several directions:

• Extension to Infinite and Lie Groups: While formulated for finite groups, foundational aspects suggest refinements to compact Lie groups given appropriate treatments of equivariant duality and descent.
• Genuine Equivariant Spectra and Mackey Functors: The pro-$\mathcal{C}$5-anima and shape-theoretic approach indicates connections with the theory of spectral Mackey functors and genuine equivariant stable homotopy theory.
• Arithmetic and Motivic Applications: Applications to equivariant motivic homotopy categories and arithmetic dualities are anticipated, extending universal properties in the presence of additional cohomological structure.

Conclusion

Bredon sheaf cohomology as defined in this work integrates the formalisms of sheaf theory, equivariant cohomology, and higher categorical algebra into a robust and universal object for the study of $\mathcal{C}$6-spaces. The strong structural and uniqueness results emphasize its canonical nature and computational viability. The synthesis of topology, $\mathcal{C}$7-theory, and homotopy-theoretic shape realized herein establishes a new standard for further exploration of equivariant invariants, their geometric interpretation, and their categorical foundations.