Decidable object in a topos ($A^2={\Delta }_A⨿D_A$)

An object is decidable when its diagonal is a complemented subobject of its square. Equivalently its internal equality is decidable. Pullback gives closure under subobjects; coordinatewise equality gives closure under finite products. For an existing coproduct, unequal summands together with the complements of the diagonal in each equal summand form the diagonal complement.

A covariant set-valued functor $F$ is a decidable object precisely when every transition map $F(u)$ is injective. The pointwise complement of its diagonal consists of unequal pairs; it is a subfunctor exactly when transition maps preserve inequality. For left M-sets, this says that every action map is injective.

In a Grothendieck topos, an object is a quotient of a decidable object when it has an epic cover by one. These objects are closed under subobjects, small coproducts, quotients and finite limits. Their union inside any fixed object gives the largest subobject of this kind, defining a coreflective subcategory. The induced idempotent comonad is left exact, so the full subcategory is a topos.