Global element

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

${\displaystyle h\colon 1\to A,}$

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).

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

• 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 ${\displaystyle P}$ can be identified with the elements of ${\displaystyle P}$. Precisely, there is a natural isomorphism ${\displaystyle (1\to P)\cong \operatorname {Forget} (P)}$ where ${\displaystyle \operatorname {Forget} }$ 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 ${\displaystyle (1\to {\mathcal {C}})\cong \operatorname {Ob} ({\mathcal {C}})}$ (where ${\displaystyle \operatorname {Ob} }$ 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 ${\displaystyle G}$, there is a unique morphism ${\displaystyle 1\to G}$ (mapping the identity to the identity of ${\displaystyle G}$). 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,² whence the global elements of a graph are its self-loops.

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

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.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.