# Global element

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{{Short description|Concept in category concept}}
In [category theory](/source/category_theory), a '''global element''' of an [object](/source/object_(category_theory)) ''A'' from a [category](/source/Category_(mathematics)) is a [morphism](/source/morphism)
:<math>h\colon 1 \to A,</math>
where {{math|1}} is a [terminal object](/source/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](/source/category_of_sets), and they can be used to import [set-theoretic](/source/set_theory) 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](/source/up_to) [isomorphism](/source/isomorphism)).

==Examples==

* In the [category of sets](/source/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](/source/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](/source/concrete_category), in the [category of partially ordered sets](/source/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](/source/forgetful_functor) from the category of posets to the category of sets. The same holds in the [category of topological spaces](/source/category_of_topological_spaces).

* Similarly, in the [category of (small) categories](/source/category_of_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](/source/objects_functor)).

* As an example where global elements do ''not'' recover elements of sets, in the [category of groups](/source/category_of_groups), the terminal objects are [zero group](/source/zero_group)s. 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](/source/zero_object) (such as the [category of abelian groups](/source/category_of_abelian_groups) or the [category of vector spaces](/source/category_of_vector_spaces) on a field), each object has a unique global element.

* In the [category of graphs](/source/category_of_graphs), the terminal objects are graphs with a single vertex and a single [self-loop](/source/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](/source/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](/source/section_(category_theory)) of <math>f</math>.

==In topos theory==

In an [elementary topos](/source/elementary_topos) the global elements of the [subobject classifier](/source/subobject_classifier) form a [Heyting algebra](/source/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](/source/Clique_(graph_theory)) 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 value](/source/truth_value)s.

==References==
{{reflist}}

==See also==

* [Well-pointed category](/source/Well-pointed_category)

Category:Category theory

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Adapted from the Wikipedia article [Global element](https://en.wikipedia.org/wiki/Global_element) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Global_element?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.