Differential spaces, diffeological spaces and fr¨olicher spaces: a comparative smootheology.

The contemporary smooth structures that describe the motion of particles on underlying sets through the related specified differential geometry need to be assessed in the sense of stabilising the advantage of working with one or another. Such a study is what we call a comparative smootheology. A diffeological structure,differential structure and a Fr¨olicher structure are each a general-isation of a smooth manifold structure. However, it is known that a smooth manifold is a Fr¨olicher space, a Fr¨olicher space is a subcategory of a diffeology and a Fr¨olicher space is a subcategory of a differential space.In this study we will carry out a comparative study on the three spaces and see to which extent the diffeologies, the differential structure in the sense of Sikorski or the Fr¨olicher structure will be more suited in describing any field of application that require the tools of differential geometry. The method of comparison will be based on comparing, their structures, their topologies and their tangent structures.