Groupoids in Sikorski's spaces, connections and the Chern-Weil homomorphism. (Q2782571)

Let \(A\) be a Lie algebroid on a differentiable manifold \(M\) with a Lie algebra structure \([[\cdot, \cdot]]\) and an anchor map \(\gamma: A\to TM\). The author extends the notion of a connection in a regular Lie algebroid to that in a non-regular Lie algebroid, that is, when Im\,\(\gamma\) is a foliation with singularities. The author generalizes the notion of a vector bundle to a vector semibundle admitting the existence of fibers of different dimensions, by going beyond the category of differentiable manifolds into the category of differential spaces of \textit{R. Sikorski} [Colloq. Math. 18, 251--272 (1967; Zbl 0162.25101)]. Then the author describes the non-regular Lie algebroid as an algebroid of a groupoid in the category of differential spaces, called F-groupoid (foliated groupoid). The Chern-Weil homomorphism of the F-groupoid is constructed and its independence of the choice of the connection in the F-groupoid is shown.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00030].