An object in a category is subterminal if any two morphisms with target but 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 morpjism is monic; hence the name “sub-terminal.”
If the product exists, it is equivalent to saying that the diagonal is an isomorphism.
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 .
Revised on June 26, 2011 15:52:41
by Urs Schreiber