In category theory, a **global element** of an object *A* from a category is a morphism

where 1 is a terminal object of the category.^{[1]} 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 (not even up to isomorphism). For example, the terminal object of the category **Grph** of graph homomorphisms has one vertex and one edge, a self-loop,^{[2]} 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 Ω form a Heyting algebra when ordered by inclusion of the corresponding subobjects of the terminal object.^{[3]} For example, **Grph** happens to be a topos, whose subobject classifier Ω is a two-vertex directed clique with an additional self-loop (so five edges, three of which are self-loops and hence the global elements of Ω). 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 *A* → *B* in the category, there should exist a global element whose compositions with them are different from each other.^{[1]}

## References

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^{a}^{b}Mac Lane, Saunders; Moerdijk, Ieke (1992),*Sheaves in geometry and logic: A first introduction to topos theory*, Universitext, New York: Springer-Verlag, p. 236, ISBN 0-387-97710-4, MR 1300636. **^**Gray, John W. (1989), "The category of sketches as a model for algebraic semantics",*Categories in computer science and logic (Boulder, CO, 1987)*, Contemp. Math.,**92**, Amer. Math. Soc., Providence, RI, pp. 109–135, doi:10.1090/conm/092/1003198, MR 1003198.**^**Nourani, Cyrus F. (2014),*A functorial model theory: Newer applications to algebraic topology, descriptive sets, and computing categories topos*, Toronto, ON: Apple Academic Press, p. 38, doi:10.1201/b16416, ISBN 978-1-926895-92-5, MR 3203114.