Simplicial sheaves of modules and Morita invariance of groupoid cohomology

2025/09/08 by Xiang Tang, Joel Villatoro, Tang, Xiang +1
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2509.07285

openalex publication_date 2025/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we develop a unified framework for proving Morita invariance of cohomology theories associated to Lie groupoids. Our approach is to view these cohomology theories as arising from sheaves of modules on the nerve of the groupoid. We establish criteria for when such sheaves of modules give rise to Morita invariant cohomology theories.

Citations

• On the cohomology of differentiable stacks
• Van Est differentiation and integration
• On the Van Est homomorphism for Lie groupoids
• Differential cohomology
• Representations up to homotopy and Bott's spectral sequence for Lie groupoids
• Differentiable Stacks and Gerbes
• Cech-De Rham theory for leaf spaces of foliations
• Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes

Related

• Modules over etale groupoid algebras as sheaves
• Morita theory for Hopf algebroids and presheaves of groupoids
• Morita invariance of equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids
• Principal bundles with groupoid structure: local vs. global theory and nonabelian ucech cohomology
• On the cohomology of differentiable stacks