On Theories of Superalgebras of Differentiable Functions

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.