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category theory

# Contents

## Definition

###### Definition

An object $U$ in a category $C$ is subterminal or preterminal 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, so that $U$ represents a subobject of $1$; hence the name “sub-terminal.”

This is equivalent to the hypothesis that the cone given by identity morphisms $U \leftarrow U \rightarrow U$ is a product cone, or that some product $U \times U$ exists and 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).

## Examples

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 $X \to U$.

## References

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

• Dieter Pumplün, Initial morphisms and monomorphisms, Manuscripta mathematica 32 (1980): 309-333.

Last revised on February 22, 2024 at 04:18:28. See the history of this page for a list of all contributions to it.