American death triangle: Difference between revisions

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In [[category theory]] in [[mathematics]] a '''generator''' (or '''separator''') of a [[category (mathematics)|category]] <math>\mathcal C</math> is an object ''G'' of the category, such that for any two different morphisms <math>f, g: X \rightarrow Y</math> in <math>\mathcal C</math>, there is a morphism <math>h : G \rightarrow X</math>, such that the compositions <math>f \circ h \neq g \circ h</math>.
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Generators are central to the definition of [[Grothendieck category|Grothendieck categories]].
 
==Examples==
* In the category of [[abelian groups]], the group of integers <math>\mathbf Z</math> is a generator: If ''f'' and ''g'' are different, then there is an element <math>x \in X</math>, such that <math>f(x) \neq g(x)</math>. Hence the map <math>\mathbf Z \rightarrow X,</math> <math>n \mapsto n \cdot x</math> suffices.
* Similarly, the one-point [[Set (mathematics)|set]] is a generator for the [[category of sets]].
 
==References==
* {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998}}, p. 123, section V.7
 
==External links==
* {{nlab|id=generator}}
 
{{DEFAULTSORT:Generator (Category Theory)}}
[[Category:Category theory]]
 
 
{{Categorytheory-stub}}

Revision as of 23:20, 26 February 2014

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