Čech cocycle groupoid

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. If $h:g\to g'$ and $k:g'\to g''$ , their composite $k\circ h$ is represented by $(h_i k_i)_i$ . The identity has every component $e$ , and the inverse of $h$ has components $h_i^{-1}$ . Associativity follows from group multiplication.

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 trivializable over $\mathcal U$ , with all bundle isomorphisms; choosing local trivializations gives the cocycle presentation. 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.