Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of differentiable groupoids and stratified groupoids, generalizations Lie in which spaces objects arrows have structures spaces, respectively compatible with groupoid structure. After studying basic properties these including Morita equivalence, we prove a de Rham theorem for proper locally contractible groupoids. then focus on study inertia associated to groupoid. show that loop space can be endowed natural Whitney (b)-regular stratification, call orbit Cartan type stratification. Endowed this becomes