  We generalize the differential space concept as a tool for developing
differential geometry, and enrich this geometry with infinitesimals that allow
us to penetrate into the superfine structure of space. This is achieved by
Yoneda embedding a ring of smooth functions into the category of loci. This
permits us to define a category of functorial differential spaces. By suitably
choosing various algebras as "stages" in this category, one obtains various
classes of differential spaces, both known from the literature and many so far
unknown. In particular, if one chooses a Weil algebra, infinitesimals are
produced. We study the case with some Weil algebra which allows us to fully
develop the corresponding differential geometry with infinitesimals. To test
the behavior of infinitesimals, we construct a simplified RWFL cosmological
model. As it should be expected, infinitesimals remain latent during the entire
macroscopic evolution (regarded backwards in time), and come into play only
when the universe attains infinitesimal dimensions. Then they penetrate into
the structure of the initial singularity.
