On theories of superalgebras of differentiable functions (Q2859013)

Fermat theories, providing a unifying framework for the algebraic study of polynomials based on commutative rings on the one hand and that of smooth functions based on \(\mathbf{C}^{\infty}\)-rings on the other, were introduced in [Zbl 1254.51005]. This paper, which is the first one of a series of papers aimed at laying the foundations for a differential graded approach to derived differential geometry, introduces super Fermat theories dealing with supercommutative algebras in which not only polynomials but also infinitely differentiable functions can be evaluated. Just as \(\mathbf{C}^{\infty}\)-rings have played a pivotal role in the development of models of synthetic differential geometry, it is \(\mathbf{C}^{\infty}\)-superalgebras that are of central importance in the authors' approach. The superization of a Fermat theory is a \(2\)-sorted algebraic theory abiding by the super Fermat property. It should be stressed that not every super Fermat theory arises as a superization. It is shown that the superization function from Fermat theories to super Fermat theories is left adjoint to the underlying functor. Near-point determined algebras for a super Fermat theory are investigated thoroughly. Two appendices are concerned with algebraic theories and multisorted Lawvere theories.