{{Short description|Concept in category concept}} In category theory, a '''global element''' of an object ''A'' from a category is a morphism :<math>h\colon 1 \to A,</math> where {{math|1}} is a terminal object of the category.<ref>{{citation | last1 = Mac Lane | first1 = Saunders | author1-link = Saunders Mac Lane | last2 = Moerdijk | first2 = Ieke | author2-link = Ieke Moerdijk | isbn = 0-387-97710-4 | location = New York | mr = 1300636 | page = 236 | publisher = Springer-Verlag | series = Universitext | title = Sheaves in geometry and logic: A first introduction to topos theory | url = https://books.google.com/books?id=SGwwDerbEowC&pg=PA236 | year = 1992}}.</ref> 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 <math>A</math> can be assimilated to an element of <math>A</math> in the usual (set-theoretic) sense. More precisely, there is a natural isomorphism <math>(1 \to A) \cong A</math>.
* 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 <math>P</math> can be identified with the elements of <math>P</math>. Precisely, there is a natural isomorphism <math>(1 \to P) \cong \operatorname{Forget}(P)</math> where <math>\operatorname{Forget}</math> 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 <math>(1 \to \mathcal{C}) \cong \operatorname{Ob}(\mathcal{C})</math> (where <math>\operatorname{Ob}</math> 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 <math>G</math>, there is a unique morphism <math>1 \to G</math> (mapping the identity to the identity of <math>G</math>). 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,<ref>{{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 }}.</ref> whence the global elements of a graph are its self-loops.
* In an overcategory <math>\mathcal{C}/B</math>, the object <math>B \overset{\operatorname{id}}{\to} B</math> is terminal. The global elements of an object <math>A \overset{f}{\to} B</math> are the sections of <math>f</math>.
==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.<ref>{{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}}.</ref> 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.
==References== {{reflist}}
==See also==
Category:Category theory