In this paper we present classifying toposes for the following theories: the theory of $\mathcal{ C}^{ \infty}-$rings, the theory of local $\mathcal{ C}^{ \infty}-$rings and the theory of von Neumann regular $\mathcal{ C}^{ \infty}-$rings. The classifying toposes for the first two theories were stated without proof by Ieke Moerdijk and Gonzalo Reyes on the page 366 of \cite{ MSIA}, where they assert that the topos ${ \rm \bf Set}^{ \mathcal{ C}^{ \infty}{ \rm \bf Rng}_{ \rm fp}}$ classifies the theory of $\mathcal{ C}^{ \infty}-$rings and that the smooth Zariski topos classifies the theory of local $\mathcal{ C}^{ \infty}-$rings.
Categories: math.LO math.CT