nLab
subterminal object

Definition

Definition

An object $U$ in a category $C$ is subterminal if any two morphisms with target $U$ but the same source are equal. In other words, $U$ is subterminal if for any object $X$, there is at most one morphism $X\to U$.

Definition

An umbrella category is a nonempty category $C$ such that for every object $X$ in $C$, there is at least one subterminal object $T$ such that $C(X,T)$ is nonempty (hence being a singleton).

Properties

If $C$ has a terminal object $1$, then $U$ is subterminal precisely if the unique morpjism $U\to 1$ is monic; hence the name “sub-terminal.”

If the product $U\times U$ exists, it is equivalent to saying that the diagonal $U\to U\times U$ is an isomorphism.

Examples

The subterminal objects in a topos can be viewed as its “external truth values.” For example, in the topos $\mathrm{Sh}(X)$ of sheaves on a topological space $X$, the subterminal objects are precisely the open sets in $X$.