Global element

In category theory, a global element of an object A from a category is a morphism
where is a terminal object of the category. Roughly speaking, global elements are a generalization of the notion of "elements" from the category of sets, and they can be used to import set-theoretic concepts into category theory. However, unlike a set, an object of a general category need not be determined by its global elements.

Examples

In the category of sets, the terminal objects are the singletons, so a global element of can be assimilated to an element of in the usual sense. More precisely, there is a natural isomorphism.
To illustrate that the notion of global elements can sometimes recover the actual elements of the objects in a concrete category, in the category of partially ordered sets, the terminal objects are again the singletons, so the global elements of a poset can be identified with the elements of. Precisely, there is a natural isomorphism where is the forgetful functor from the category of posets to the category of sets. The same holds in the category of topological spaces.
Similarly, in the category of (small) categories, terminals objects are unit categories. Consequently, a global element of a category is simply an object of that category. More precisely, there is a natural isomorphism .
As an example where global elements do not recover elements of sets, in the category of groups, the terminal objects are zero groups. For any group, there is a unique morphism . More generally, in any category with a zero object, each object has a unique global element.
In the category of graphs, the terminal objects are graphs with a single vertex and a single self-loop on that vertex, whence the global elements of a graph are its self-loops.
In an overcategory, the object is terminal. The global elements of an object are the sections of.

In an elementary topos the global elements of the subobject classifier form a Heyting algebra when ordered by inclusion of the corresponding subobjects of the terminal object. For example, Grph happens to be a topos, whose subobject classifier is a two-vertex directed clique with an additional self-loop. The internal logic of Grph is therefore based on the three-element Heyting algebra as its truth values.