  In this paper, we use some basic quasi-topos theory to study two functors:
one adding infinitesimals of Fermat reals to diffeological spaces (which
generalize smooth manifolds including singular spaces and infinite dimensional
spaces), and the other deleting infinitesimals on Fermat spaces. We study the
properties of these functors, and calculate some examples. These serve as
fundamentals for developing differential geometry on diffeological spaces using
infinitesimals in a future paper.
