Continuous family groupoids

In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant Atiyah-Singer index theorem for families, and is also required in noncommutative geometry. The class includes that of Lie groupoids, and the paper shows that, like Lie groupoids, continuous family groupoids always admit (an essentially unique) continuous left Haar system of smooth measures. We also show that the action of a continuous family groupoid $G$ on a continuous family $G$-space fibered over another continuous family $G$-space $Y$ can always be regarded as an action of the continuous family groupoid $GY$ on an ordinary $GY$-space.

💡 Research Summary

The paper introduces the notion of a continuous family groupoid, a natural generalisation of Lie groupoids that is designed to handle families of groupoids parametrised by a base space. A continuous family groupoid consists of a smooth family ({G_x}_{x\in M}) of Lie groupoids indexed by points of a smooth manifold (M); the source, target, multiplication, inverse and unit maps vary continuously (indeed smoothly) with the parameter (x). In the special case where the base consists of a single point, the definition reduces to that of an ordinary Lie groupoid, showing that the new class truly extends the classical theory.

The first major result is the existence and essential uniqueness of a continuous left Haar system on any continuous family groupoid. Since each fibre (G_x) already carries a Haar measure (as a Lie groupoid), the authors construct a global system by selecting Haar measures in a fibrewise smooth manner. Using local charts, partitions of unity and a careful analysis of the dependence of the Haar density on the base variable, they prove that the resulting family of measures is continuous with respect to the groupoid topology and that any two such systems that are pointwise equivalent differ only by a continuous positive function on the base. This essentially unique Haar system is crucial for defining the convolution algebra of the groupoid, its C*-completion, and for developing representation theory.

The second key theorem concerns actions. If a continuous family groupoid (G) acts on a continuous family (G)-space (Y) that itself fibres over another (G)-space (X), the action can be re‑interpreted as an ordinary action of the “action groupoid’’ (GY) on a standard (GY)-space. The construction of (GY) as a fibred product ({(g,y)\mid s(g)=p(y)}) yields again a continuous family groupoid, and the authors verify that the Haar system on (GY) coincides with the one induced from (G). This reformulation simplifies the treatment of equivariant structures and allows the direct application of existing tools from Lie groupoid theory (e.g., Morita equivalence, groupoid C*-algebras) to more intricate family actions.

The motivation for introducing continuous family groupoids lies in two prominent areas. In the family version of the Atiyah–Singer index theorem, one studies a smooth family of elliptic operators parametrised by a base manifold; the index lives in K‑theory of a suitable C*-algebra. By modelling the family of operators as elements of the convolution algebra of a continuous family groupoid, one obtains a natural framework in which the index map can be expressed and studied. The existence of a Haar system guarantees that the convolution algebra is well‑behaved, while the action groupoid construction handles equivariant situations.

In non‑commutative geometry, groupoid C*-algebras serve as prototypes for “non‑commutative spaces”. When the underlying geometric data varies continuously, the continuous family groupoid provides the appropriate object: its C*-algebra encodes both the fibrewise non‑commutative structure and the smooth variation across the base. This makes it possible to apply K‑theoretic and cyclic cohomology techniques to families of non‑commutative spaces, extending many results that were previously limited to a single groupoid.

Overall, the paper establishes a robust categorical setting that unifies Lie groupoids, families of groupoids, and their actions. By proving the existence of a (essentially unique) continuous Haar system and by showing how actions can be encoded via the action groupoid (GY), the authors lay the groundwork for a systematic development of family index theory and for applications in non‑commutative geometry. The results open the door to further investigations, such as the study of longitudinal pseudodifferential calculi on continuous family groupoids, Morita invariance of their C-algebras, and the formulation of higher index theorems for parametrised families.