Global element

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In category theory, a global element of an object A from a category is a morphism

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where 1 is a terminal object of the category. [1] 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 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. [3] 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.

References

  1. Mac Lane, Saunders; Moerdijk, Ieke (1992), Sheaves in geometry and logic: A first introduction to topos theory, Universitext, New York: Springer-Verlag, p. 236, ISBN   0-387-97710-4, MR   1300636 .
  2. Gray, John W. (1989), "The category of sketches as a model for algebraic semantics", Categories in computer science and logic (Boulder, CO, 1987), Contemp. Math., vol. 92, Amer. Math. Soc., Providence, RI, pp. 109–135, doi:10.1090/conm/092/1003198, ISBN   978-0-8218-5100-5, MR   1003198 .
  3. Nourani, Cyrus F. (2014), A functorial model theory: Newer applications to algebraic topology, descriptive sets, and computing categories topos, Toronto, ON: Apple Academic Press, p. 38, doi:10.1201/b16416, ISBN   978-1-926895-92-5, MR   3203114 .

See also