Examples of Groupoid

In the present paper, we introduce spin groupoids associated with the degree two Stiefel-Whitney classes, and we also introduce spinor torsors equipped with certain connections compatible with an action of spin groupoid. Using the connection, we propose a Dirac-like operator acting on the space of smooth sections of the spinor torsor. We can show that its Dirac-Laplacian admits a heat kernel in a formal sense.