  Let ${\cal E}$ be a topos, ${{\rm Dec}({\cal E}) \rightarrow {\cal E}}$ be
the full subcategory of decidable objects, and ${{\cal E}_{\neg\neg}
\rightarrow {\cal E}}$ be the full subcategory of double-negation sheaves. We
give sufficient conditions for the existence of a Unity and Identity ${{\cal E}
\rightarrow {\cal S}}$ for the two subcategories of E above, making them
Adjointly Opposite. Typical examples of such ${\cal E}$ 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.
