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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.
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 value]]s.
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_(graph_theory)]] 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 value]]s.


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{{cattheory-stub}}

Revision as of 10:15, 31 March 2013

In category theory, a global element of an object A from a category 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, 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_(graph_theory) 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.