In category theory, a global element of an object A from a category is a morphism
h : 1 → A , {\displaystyle h\colon 1\to A,}
where 1 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 (not even up to isomorphism).
Examples In the category of sets, the terminal objects are the singletons, so a global element of A {\displaystyle A} can be assimilated to an element of A {\displaystyle A} in the usual (set-theoretic) sense. More precisely, there is a natural isomorphism ( 1 → A ) ≅ A {\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 P {\displaystyle P} can be identified with the elements of P {\displaystyle P} . Precisely, there is a natural isomorphism ( 1 → P ) ≅ Forget ( P ) {\displaystyle (1\to P)\cong \operatorname {Forget} (P)} where Forget {\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 ( 1 → C ) ≅ Ob ( C ) {\displaystyle (1\to {\mathcal {C}})\cong \operatorname {Ob} ({\mathcal {C}})} (where Ob {\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 G {\displaystyle G} , there is a unique morphism 1 → G {\displaystyle 1\to G} (mapping the identity to the identity of G {\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, whence the global elements of a graph are its self-loops. In an overcategory C / B {\displaystyle {\mathcal {C}}/B} , the object B → id B {\displaystyle B{\overset {\operatorname {id} }{\to }}B} is terminal. The global elements of an object A → f B {\displaystyle A{\overset {f}{\to }}B} are the sections of f {\displaystyle f} .
In topos theory 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 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.
References
See also Well-pointed category
