Global element
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).
Examples
[edit]- In the category of sets, the terminal objects are the singletons, so a global element of can be assimilated to an element of in the usual (set-theoretic) sense. More precisely, there is a natural isomorphism .
- 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 can be identified with the elements of . Precisely, there is a natural isomorphism where 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 (where 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 , there is a unique morphism (mapping the identity to the identity of ). 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 , the object is terminal. The global elements of an object are the sections of .
In topos theory
[edit]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
[edit]- ^ 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., vol. 92, Amer. Math. Soc., Providence, RI, pp. 109–135, doi:10.1090/conm/092/1003198, ISBN 978-0-8218-5100-5, 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.