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 $X\to U$.

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).

If $C$ has a terminal object $1$, then $U$ is subterminal precisely if the unique morphism $U \to 1$ is monic, so that $U$ represents a subobject of $1$; 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.

Therefore for a sheaf topos over a topological space the subterminal objects of the topos are the open subsets of the topological space. Accordingly, the subterminal objects in any topos are also called *open objects* (e.g. Johnstone 77, p. 94)

The classifying topos for subterminal objects (hence open objects) in toposes is the Sierpinski topos (see e.g. Johnstone 77, p. 117).

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 $X \to U$.

**computational trinitarianism** =

**propositions as types** +**programs as proofs** +**relation type theory/category theory**

- Peter Johnstone,
*Topos theory*, London Math. Soc. Monographs**10**, Acad. Press 1977

Last revised on August 2, 2023 at 17:45:05. See the history of this page for a list of all contributions to it.