Preprint Nr 1, May 1986, Institute of Mathematics, Technical University of £ód¼.
publications
This is the first paper of the series of the author's works devoted to applications of the theory of differential spaces in the sense of Sikorski to groupoids. Much more general than differential groupoid, the notion of a smooth groupoid over a -manifold is defined here. Next, J.Pradines' idea of constructing, for each differential groupoid, some vector bundle with natural algebraic structures (called an algebroid of this diff. groupoid) -- playing the analogous role to that of a Lie algebra of a Lie group -- is used for smooth groupoids. Namely, some object, more general than a vector bundle, is assigned to each smooth groupoid. A particularly interesting situation takes place in the case when, for a given smooth groupoid, this object is a vector bundle. The groupoid is then called a groupoid of Pradines type. There are much more groupoids of this type than differential ones. For example, the equivalence relation of any (not only simple) foliation is such a groupoid. A large class of other examples is given. The idea of Pradines-type groupoids with singularities, inspired by foliations with singularities in the sense of P.Stefan, and the Frobenius-Sussmann theorem, -- is given. Some new facts from the theory of differential spaces we shall need are obtained.