Classifying toposes for some theories of $\mathcal{C}^{\infty}-$rings

Published 21 Nov 2018 in math.LO and math.CT | (1811.08838v1)

Abstract: 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.}