Čech cocycle groupoid

Definition

Fix an open cover $\mathcal U=\{U_i\}$ and a Lie group $G$ . The Čech cocycle groupoid has smooth $G$ -valued Čech $1$ -cocycles $g=\{g_{ij}\}$ as objects. A morphism $g\to g'$ is a family of smooth maps $h_i:U_i\to G$ such that

g'_{ij}=h_i^{-1}g_{ij}h_j

on every overlap. Composition is induced by pointwise multiplication of the families $\{h_i\}$ , with order determined by this convention.

Why it is a groupoid

Every morphism is invertible because each $h_i$ is group-valued. Passing to isomorphism classes gives the pointed set of nonabelian Čech $H^1$ , while retaining the groupoid also retains the automorphisms of each cocycle. For a cocycle defining a principal bundle, these automorphisms correspond to its gauge transformations.

Relation to bundle classification

The gluing construction identifies this groupoid with the groupoid of principal $G$ -bundles equipped with trivializations over $\mathcal U$ . Refining covers gives the cover-independent descent description.

References

Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser, 1993. DOI record. Relevant: Čech cocycles, descent, and bundle classification.