A Universal Algebraic Survey of $\mathcal{C}^{\infty}-$Rings

Abstract: In this paper we present some basic results of the Universal Algebra of $\mathcal{C}^\infty$-rings which were nowhere to be found in the current literature. The outstanding book of I. Moerdijk and G. Reyes,[24], presents the basic (and advanced) facts about $\mathcal{C}^\infty$-rings, however such a presentation has no universal algebraic "flavour". We have been inspired to describe $\mathcal{C}^\infty$-rings through this viewpoint by D. Joyce in [15]. Our main goal here is to provide a comprehensive material with detailed proofs of many known "taken for granted" results and constructions used in the literature about $\mathcal{C}^\infty$-rings and their applications - such proofs either could not be found or were merely sketched. We present, in detail, the main constructions one can perform within this category, such as limits, products, homomorphic images, quotients, directed colimits, free objects and others, providing a "propaedeutic exposition" for the reader's benefit.