Using the language of category theory, a Lie groupoid can be more compactly defined as a groupoid (i.e. a small category where all the morphisms are invertible) such that the sets M of objects and G of morphisms are manifolds, the maps s , t , m , i and u are smooth and s and t are submersions. A Lie groupoid is therefore not simply a groupoid object in the category of smooth manifolds : one has to ask the additional property that s and t are submersions.