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- Global_element abstract "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 graph homomorphisms 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.A well-pointed category is a category that has enough global elements to distinguish every two arrows. That is, for each two different arrows A → B in the category, there should exist a global element whose compositions with them are different from each other.".
- Global_element wikiPageID "14717987".
- Global_element wikiPageLength "3118".
- Global_element wikiPageOutDegree "13".
- Global_element wikiPageRevisionID "621143078".
- Global_element wikiPageWikiLink Category:Objects_(category_theory).
- Global_element wikiPageWikiLink Category_(mathematics).
- Global_element wikiPageWikiLink Category_of_sets.
- Global_element wikiPageWikiLink Category_theory.
- Global_element wikiPageWikiLink Clique_(graph_theory).
- Global_element wikiPageWikiLink Graph_homomorphism.
- Global_element wikiPageWikiLink Initial_and_terminal_objects.
- Global_element wikiPageWikiLink Isomorphism.
- Global_element wikiPageWikiLink Subobject_classifier.
- Global_element wikiPageWikiLink Topos.
- Global_element wikiPageWikiLink Truth_value.
- Global_element wikiPageWikiLink Up_to.
- Global_element wikiPageWikiLink Well-pointed_category.
- Global_element wikiPageWikiLinkText "global element".
- Global_element wikiPageUsesTemplate Template:Cattheory-stub.
- Global_element wikiPageUsesTemplate Template:Reflist.
- Global_element subject Category:Objects_(category_theory).
- Global_element hypernym H.
- Global_element type ChemicalCompound.
- Global_element comment "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).".
- Global_element label "Global element".
- Global_element sameAs Q5570834.
- Global_element sameAs m.03gvb28.
- Global_element sameAs Q5570834.
- Global_element wasDerivedFrom Global_element?oldid=621143078.
- Global_element isPrimaryTopicOf Global_element.