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generator of a category
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(Definition)
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Let
 be a category, and
 a pair of distinct morphisms. A morphism  is said to distinguish or separate  and  if
 . For example, if
 , then  on  distinguishes  and  .
A set
 of objects (indexed by a set  ) is called a generating set of
 if any pair of distinct morphisms
 can be distinguished by a morphism with domain in  and codomain  . In other words, there is
 for some  , such that
 . If
 is a generating family of
 , then  is called a generator of
 . Any set of morphisms containing a generator is a generating set.
Examples
- In Set, the category of sets, any singleton is a generator. Suppose
 are distinct functions, so that
 for some  . Let
 be any singleton. Then
 defined by  is the function distinguishing  and  : for
 .
- In Rng, the category of rings, the ring
 is a generator. If
 are distinct ring homomorphisms, say,
 for some  . Then the ring homomorphism
 given by  distinguishes  and  .
Remark. A projective object that is also a generator is called a progenerator.
- 1
- F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)
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"generator of a category" is owned by CWoo.
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Cross-references: projective object, ring homomorphisms, rings, functions, singleton, category of sets, generating, codomain, domain, indexed by, objects, morphisms, category
There are 14 references to this entry.
This is version 7 of generator of a category, born on 2008-09-03, modified 2008-09-22.
Object id is 10987, canonical name is GeneratorOfACategory.
Accessed 484 times total.
Classification:
| AMS MSC: | 18A99 (Category theory; homological algebra :: General theory of categories and functors :: Miscellaneous) |
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Pending Errata and Addenda
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