global element

• In the category of sets, the terminal objects are the singletons, so a global element of $A$ can be assimilated to an element of $A$ in the usual (set-theoretic) sense. More precisely, there is a natural isomorphism $(1\; \backslash to\; A)\; \backslash cong\; A$.

• 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 $P$ can be identified with the elements of $P$. Precisely, there is a natural isomorphism $(1\; \backslash to\; P)\; \backslash cong\; \backslash operatorname\{Forget\}(P)$ where $\backslash operatorname\{Forget\}$ 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 (having a single object and a single morphism which is the identity of that object). Consequently, a global element of a category is simply an object of that category. More precisely, there is a natural isomorphism $(1\; \backslash to\; \backslash mathcal\{C\})\; \backslash cong\; \backslash operatorname\{Ob\}(\backslash mathcal\{C\})$ (where $\backslash operatorname\{Ob\}$ is the objects functor).

• 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 $G$, there is a unique morphism $1\; \backslash to\; G$ (mapping the identity to the identity of $G$). More generally, in any category with a zero object (such as the category of abelian groups or the category of vector spaces on a field), 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,{{citation

| last = Gray | first = John W.

| contribution = The category of sketches as a model for algebraic semantics

| doi = 10.1090/conm/092/1003198

| mr = 1003198

| pages = 109–135

| publisher = Amer. Math. Soc., Providence, RI

| series = Contemp. Math.

| title = Categories in computer science and logic (Boulder, CO, 1987)

| url = https://books.google.com/books?id=boJYH2nIX6oC&pg=PA114

| volume = 92

| year = 1989| isbn = 978-0-8218-5100-5

}}. whence the global elements of a graph are its self-loops.

• In an overcategory $\backslash mathcal\{C\}/B$, the object $B\; \backslash overset\{\backslash operatorname\{id\}\}\{\backslash to\}\; B$ is terminal. The global elements of an object $A\; \backslash overset\{f\}\{\backslash to\}\; B$ are the sections of $f$.

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.{{citation

| last = Nourani | first = Cyrus F.

| doi = 10.1201/b16416

| isbn = 978-1-926895-92-5

| location = Toronto, ON

| mr = 3203114

| page = 38

| publisher = Apple Academic Press

| title = A functorial model theory: Newer applications to algebraic topology, descriptive sets, and computing categories topos

| url = https://books.google.com/books?id=v6CNAgAAQBAJ&pg=PA38

| year = 2014}}. For example, Grph happens to be a topos, whose subobject classifier {{math|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 {{math|Ω}}). The internal logic of Grph is therefore based on the three-element Heyting algebra as its truth values.

{{reflist}}

• Well-pointed category

Category:Category theory