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. For example, the terminal object of the category Grph of graph homomorphisms has one vertex and one edge, a self-loop, whence the global elements of a graph are its self-loops, conveying no information either about other kinds of edges, or about vertices having no self-loop, or about whether two self-loops share a vertex.
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.
A well-pointed category is a category that has enough global elements to distinguish every two arrows. That is, for each pair of distinct arrows in the category, there should exist a global element whose compositions with them are different from each other.