Derived $C^{\infty}$-Geometry I: Foundations

This work is the first in a series laying the foundations of derived geometry in the $C^{\infty}$ setting, and providing tools for the construction and study of moduli spaces of solutions of Partial Differential Equations that arise in differential geometry and mathematical physics. To advertise the advantages of such a theory, we start with a detailed introduction to derived $C^{\infty}$-geometry in the context of symplectic topology and compare and contrast with Kuranishi space theory. In the body of this work, we avail ourselves of Lurie's extensive work on abstract structured spaces to define $\infty$-categories of derived $C^{\infty}$-rings and $C^{\infty}$-schemes and derived $C^{\infty}$-rings and $C^{\infty}$-schemes with corners via a universal property in a suitable $(\infty,2)$-category of $\infty$-categories with respect to the ordinary categories of manifolds and manifolds with corners (with morphisms the $b$-maps of Melrose in the latter case), and prove many basic structural features about them. Along the way, we establish some derived flatness results for derived $C^{\infty}$-rings of independent interest.