On groupoids with involutions and their cohomology

We extend the definitions and main properties of graded extensions to the category of locally compact groupoids endowed with involutions. We introduce Real \v{C}ech cohomology, which is an equivariant-like cohomology theory suitable for the context of groupoids with involutions. The Picard group of such a groupoid is discussed and is given a cohomological picture. Eventually, we generalize Crainic’s result, about the differential cohomology of a proper Lie groupoid with coefficients in a given representation, to the topological case.

💡 Research Summary

The paper introduces a new categorical framework—real (or “involutive”) groupoids—by equipping a locally compact groupoid with an involution τ that squares to the identity and acts compatibly on objects and arrows. After setting up this notion, the authors develop a cohomology theory tailored to such structures, called Real Čech cohomology. The construction mirrors the classical Čech cohomology of a groupoid: one chooses an open cover of the unit space, forms the usual Čech cochain complex with coefficients in a τ‑equivariant sheaf (for example the sheaf of continuous U(1)‑valued functions), and then restricts to the τ‑invariant subcomplex. The resulting cohomology groups Hⁿ_R(G, 𝔄) capture precisely the information that is invariant under the involution, reducing to ordinary Čech cohomology when τ is trivial and providing genuinely new invariants when τ is non‑trivial.

With this cohomology in hand, the authors turn to the Picard group of a real groupoid. The Picard group classifies real line bundles (or, equivalently, U(1)‑principal bundles equipped with a compatible τ‑action) over the groupoid. By interpreting a real line bundle as a global section of the τ‑equivariant sheaf 𝔘(1)_R, they prove a natural isomorphism \