An object$U$ in a category$C$ is subterminal if any two morphisms with target$U$ and 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 morphism $U \to 1$ 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.

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

The support of an object $X$ in a topos is the image$U \hookrightarrow 1$ of the unique map $X \to 1$. Any map $U \to X$ is necessarily a section of $X \to U$.