Connectedness through decidable quotients
Enrique Ruiz Hernández, Pedro Solórzano
TL;DR
This work investigates how topos-theoretic connectedness can be detected via decidable quotients by introducing the DQO axiom (there exists a unique decidable quotient through which every arrow to the truth object factors) and studying its interaction with McLarty's SDG framework. Under NS (Nullstellensatz) and DQO, the decidable objects form an exponential-ideal and the left adjoint Π to the inclusion Dec(E) ↪ E exists, providing a robust notion of connectedness with Pi(X) = 1 characterizing connected X. A central technical tool is the Fiber Pneumoconnectedness Lemma, which equates several conditions about epimorphisms with pneumoconnected fibers and the factorization property for maps to decidable targets, yielding product-preservation and topos-structure results for Dec(E). The paper then proves a converse: when E is precohesive over Dec(E) (under NS and DQO), the SDG-style axioms (DSO) hold, and conversely, in a boolean base, precohesiveness implies DQO, thereby tying connectedness, decidable quotients, and precohesion into a coherent framework with potential applications to precohesive toposes and SDG. Overall, the work extends McLarty’s program by integrating decidable quotients into a precohesive setting and clarifying when connectedness in this context matches a topos-theoretic left adjoint image.
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 $ΠX=1$, where $Π$ 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.