subterminal object





An object UU in a category CC is subterminal if any two morphisms with target UU and the same source are equal. In other words, UU is subterminal if for any object XX, there is at most one morphism XUX\to U.


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


If CC has a terminal object 11, then UU is subterminal precisely if the unique morphism U1U \to 1 is monic; hence the name “sub-terminal.”

If the product U×UU \times U exists, it is equivalent to saying that the diagonal UU×UU \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)Sh(X) of sheaves on a topological space XX, the subterminal objects are precisely the open sets in XX.

The support of an object XX in a topos is the image U1U \hookrightarrow 1 of the unique map X1X \to 1. Any map UXU \to X is necessarily a section of XUX \to U.


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

Last revised on July 22, 2016 at 13:07:54. See the history of this page for a list of all contributions to it.