We define Weil spaces, Weil manifolds, Weil varieties and Weil Lie groups
over an arbitrary commutative base ring K (in particular, over discrete rings
such as the integers), and we develop the basic theory of such spaces, leading
up the definition of a Lie algebra attached to a Weil Lie group. By definition,
the category of Weil spaces is the category of functors from K-Weil algebras to
sets; thus our notion of Weil space is similar to, but weaker than the one of
Weil topos defined by E. Dubuc (1979). In view of recent result on Weil
functors for manifolds over general topological base fields or rings by A.
Souvay, this generality is the suitable context to formulate and to prove
general results of infinitesimal differential geometry, as started by the
approach developed in Bertram, Mem. AMS 900