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>.
Hi there, I am Sophia. I've always loved living in Kentucky but now I'm contemplating other options. The preferred pastime for him and his kids is to perform lacross and he'll be starting something else alongside with it. Since he was eighteen he's been working as an information officer but he plans on changing it.<br><br>My blog; clairvoyants; [http://black7.mireene.com/aqw/5741 mireene.com],
 
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}}

Latest revision as of 01:55, 9 October 2014

Hi there, I am Sophia. I've always loved living in Kentucky but now I'm contemplating other options. The preferred pastime for him and his kids is to perform lacross and he'll be starting something else alongside with it. Since he was eighteen he's been working as an information officer but he plans on changing it.

My blog; clairvoyants; mireene.com,

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