Cantor's theorem: Difference between revisions

Revision as of 19:53, 21 November 2023

Revision as of 01:01, 22 November 2023

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==Generalizations==

Cantor's theorem has been generalized to any [[Category (mathematics)|category]] with [[Product (category theory)|products]] in the following way:<ref name="LawvereSchanuel2009">{{cite book|author1=F. William Lawvere|author2=Stephen H. Schanuel|title=Conceptual Mathematics: A First Introduction to Categories|year=2009|publisher=Cambridge University Press|isbn=978-0-521-89485-2|at=Session 29|url-access=registration|url=https://archive.org/details/conceptualmathem00lawv}}</ref> suppose that <math>Y</math> is an object in a category <math>C</math> with terminal object <math>1</math>, and there exists an endomorphism <math>\alpha : Y \to Y</math> that does not have any fixed points; that is, for every morphism <math>xy:1 \to Y</math>, <math>\alpha \circ xy \ne xy</math>. Then there is no object <math>T</math> of <math>C</math> such that a morphism <math>f: T \times T \to Y</math> can parameterize all morphisms <math>T \to Y</math>. In other words, for every object <math>T</math> and every morphism <math>f : T \times T \to Y</math>, an attempt to write maps <math>T \to Y</math> as maps of the form <math>f(-,x) : T \times T \to Y</math> must leave out at least one map <math>T \to Y</math>.

==See also==