A decidable object is the categorical rendering of the notion of a decidable set from computability theory. This corresponds to the algebraic and topological concepts separable , respectively unramified object, as pointed out by Lawvere.
For an object this means that in the internal logic of the category, it is true that “for any , either or ”.
and are always decidable, and so is every natural numbers object in a topos. A subobject of a decidable object is decidable.
Decidable maps in the opposite of the category of commutative rings are precisely the separable -algebras .
Of course, in a Boolean category, every object is decidable. Conversely in a topos , or more generally a coherent category with a subobject classifier, every object is decidable precisely if is Boolean.
An object in a topos is called anti-decidable if in the internal language of holds for all . A formula is called almost decidable iff holds and an object is called almost decidable if is almost decidable for .
B. P. Chisala, M.-M. Mawanda, Counting Measure for Kuratowski Finite Parts and Decidability , Cah.Top.Géom.Diff.Cat. XXXII 4 (1991) pp.345-353. (pdf)
Peter Johnstone, Sketches of an Elephant vols. I,II, Oxford UP 2002.