The Unity and Identity of decidable objects and double-negation sheaves

Abstract

Let $\mathcal{E}$ be a topos, $\mathrm{Dec}(\mathcal{E})\to \mathcal{E}$ be the full subcategory of decidable objects, and ${\mathcal{E}}_{\neg \neg }\to \mathcal{E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity $\mathcal{E}\to \mathcal{S}$ for the two subcategories of E above, making them Adjointly Opposite. Typical examples of such $\mathcal{E}$ include many `gros' toposes in Algebraic Geometry, simplicial sets and other toposes of `combinatorial' spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.