The volume develops the foundations of differential geometry so
as to include finite-dimensional spaces with singularities and
nilpotent functions, at the same level as is standard in the
elementary theory of schemes and analytic spaces. The theory of
differentiable spaces is developed to the point of providing a
handy tool including arbitrary base changes (hence fibred products,
intersections and fibres of morphisms), infinitesimal
neighbourhoods, sheaves of relative differentials, quotients by
actions of compact Lie groups and a theory of sheaves of Frechet
modules paralleling the useful theory of quasi-coherent sheaves on
schemes. These notes fit naturally in the theory of C
DEGREES\infinity-rings and C DEGREES\infinity-schemes, as well as
in the framework of Spallek's C DEGREES\infinity-standard
differentiable spaces, and they require a certain familiarity with
commutative algebra, sheaf theory, rings of differentiable
functions and Frechet spaces."
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