  This is the first in a series of papers laying the foundations for a
differential graded approach to derived differential geometry (and other
geometries in characteristic zero). In this paper, we study theories of
supercommutative algebras for which infinitely differentiable functions can be
evaluated on elements. Such a theory is called a super Fermat theory. Any
category of superspaces and smooth functions has an associated such theory.
This includes both real and complex supermanifolds, as well as algebraic
superschemes. In particular, there is a super Fermat theory of C-infinity
superalgebras. C-infinity superalgebras are the appropriate notion of
supercommutative algebras in the world of C-infinity rings, the latter being of
central importance both to synthetic differential geometry and to all existing
models of derived smooth manifolds. A super Fermat theory is a natural
generalization of the concept of a Fermat theory introduced by E. Dubuc and A.
Kock. We show that any Fermat theory admits a canonical superization, however
not every super Fermat theory arises in this way. For a fixed super Fermat
theory, we go on to study a special subcategory of algebras called near-point
determined algebras, and derive many of their algebraic properties.
