Connectedness through decidable quotients

Abstract

By looking at decidable quotients, a sufficient condition is provided to guarantee that (1) the full subcategory of decidable objects of a topos is an exponential ideal and that (2) the classical notion of connectedness for an object $X$ coincides with $\Pi X=1$, where $\Pi$ is the left-adjoint functor of the inclusion of the decidable objects. The addition of this condition to McLarty's axiomatic set up for Synthetic Differential Geometry makes any topos that satisfies it precohesive over the topos of its decidable objects. A converse is also provided.