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

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).

Examples

  • 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 \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 can be identified with the elements of P. Precisely, there is a natural isomorphism (1 \to P) \cong \operatorname{Forget}(P) where \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 \to \mathcal{C}) \cong \operatorname{Ob}(\mathcal{C}) (where \operatorname{Ob} is the objects functor).
  • 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 \mathcal{C}/B, the object B \overset{\operatorname{id}}{\to} B is terminal. The global elements of an object A \overset{f}{\to} B are the sections of 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.[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.

References

  1. ^ 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.
  2. ^ Gray, John W. (1989), "The category of sketches as a model for algebraic semantics", Categories in computer science and logic (Boulder, CO, 1987), Vol. 92, Contemp. Math., Amer. Math. Soc., Providence, RI, pp. 109–135, doi:10.1090/conm/092/1003198. ISBN 978-0-8218-5100-5. MR 1003198.
  3. ^ 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.

See also