Some algebraic and logical aspects of C&#8734-Rings - Berni, Jean Cerqueira (2018)
As pointed out by I. Moerdijk and G. Reyes in [63], C&#8734-rings have been studied specially for their use in Singularity Theory and in order to construct topos models for Synthetic Differential Geometry. In this work, we follow a complementary trail, deepening our knowledge about them through a more pure bias, making use of Category Theory and accounting them from a logical-categorial viewpoint. We begin by giving a comprehensive systematization of the fundamental facts of the (equational) theory of C&#8734-rings, widespread here and there in the current literature - mostly without proof - which underly the theory of C&#8734-rings. Next we develop some topics of what we call a &#8734Commutative Algebra, expanding some partial results of [66] and [67]. We make a systematic study of von Neumann-regular C&#8734-rings (following [2]) and we present some interesting results about them, together with their (functorial) relationship with Boolean spaces. We study some sheaf theoretic notions on C&#8734-rings, such as &#8734(locally)-ringed spaces and the smooth Zariski site. Finally we describe classifying toposes for the (algebraic) theory of &#8734 rings, the (coherent) theory of local C&#8734-rings and the (algebraic) theory of von Neumann regular C&#8734-rings.