subterminal

Subterminal is a term used in category theory to denote a specific kind of object relative to a terminal object. In a category that has a terminal object 1, an object A is called subterminal if there exists a monomorphism from A into 1. Equivalently, A is a subobject of the terminal object, i.e., A is classified by a subobject of 1.

In topos theory, subterminal objects play a central role because the subobjects of 1 form a lattice that encodes internal logic. The set of subterminal objects, under subobject inclusion, is a Heyting algebra, and in a topos this lattice is isomorphic to the collection of truth values used by the internal logic. Each subterminal object corresponds to a characteristic morphism into 1, serving as a truth predicate for a subobject.

Examples help illustrate the concept. In the category of Sets, the terminal object is a singleton set, and the subobjects of 1 are only the empty set and 1 itself, yielding a two-element lattice that corresponds to false and true. In other categories, especially in sheaf toposes, the lattice of subterminal objects can be richer: for the category of sheaves on a space, subterminal objects correspond to open subsets of the space, reflecting a more nuanced notion of truth.

Related notions include the subobject classifier, which generalizes the idea of a truth-value object 1 to determine subobjects via characteristic morphisms. Subterminal objects thus provide a foundational link between object-level structure and logical truth values in categorical settings.

See also: terminal object, subobject, subobject classifier, topos theory, internal logic.