nLab
subterminal object

Contents

Definition

Definition

An object UU in a category CC is subterminal if any two morphisms with target UU but 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.

Definition

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).

Properties

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.

Examples

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.

Revised on February 17, 2014 01:50:15 by Cale Gibbard? (99.247.222.118)