Global element
Encyclopedia
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. 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
). For example the terminal object of the category Grph of graphs has one vertex and one edge, a self-loop, whence the global elements of a graph are its self-loops, conveying no information either about other kinds of edges, or about vertices having no self-loop, or about whether two self-loops share a vertex.
In an elementary topos the global elements of the subobject classifier
Ω 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.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a global element of an object A from a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
is a morphism
- h : 1 → 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
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
, 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
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
). For example the terminal object of the category Grph of graphs has one vertex and one edge, a self-loop, whence the global elements of a graph are its self-loops, conveying no information either about other kinds of edges, or about vertices having no self-loop, or about whether two self-loops share a vertex.
In an elementary topos the global elements of the subobject classifier
Subobject classifier
In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements...
Ω 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
Clique
A clique is an exclusive group of people who share common interests, views, purposes, patterns of behavior, or ethnicity. A clique as a reference group can be either normative or comparative. Membership in a clique is typically exclusive, and qualifications for membership may be social or...
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.