An object in a category is subterminal if any two morphisms with target and the same source are equal. In other words, is subterminal if for any object , there is at most one morphism .
An umbrella category is a nonempty category such that for every object in , there is at least one subterminal object such that is nonempty (hence being a singleton).
If has a terminal object , then is subterminal precisely if the unique morphism is monic; hence the name “sub-terminal.”
If the product exists, it is equivalent to saying that the diagonal 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 of sheaves on a topological space , the subterminal objects are precisely the open sets in .
The support of an object in a topos is the image of the unique map . Any map is necessarily a section of .
- Peter Johnstone, Topos theory, London Math. Soc. Monographs 10, Acad. Press 1977