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enough projectives
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(Definition)
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Let $\mathcal{A}$ be an abelian category. $\mathcal{A}$ is said to have enough projectives if, for every object $A$ of $\mathcal{A}$ , there is a projective object $P$ of $\mathcal{A}$ and an exact sequence
In other words, the map $p\colon P \to A$ is epi, or an epimorphism.
Example. Let $R$ be a ring. The category of left (right) $R$ -modules is an abelian category having enough projectives. This is true since, for every left (right) $R$ -module $M$ , we can take $F$ to be the free (and hence projective) $R$ -module generated by a generating set $X$ for $M$ (we can in fact take $X$ to be $M$ ). Then the canonical projection $\pi\colon F\to M$ is the required surjection.
More generally, a category $\mathcal{C}$ is said to have enough projectives if every object is a strong quotient object of a projective object.
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- F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)
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"enough projectives" is owned by CWoo.
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Cross-references: quotient object, strong, surjection, canonical projection, generating set, generated by, right, category, ring, epimorphism, epi, map, exact sequence, projective object, object, abelian category
There are 4 references to this entry.
This is version 7 of enough projectives, born on 2004-11-21, modified 2008-09-22.
Object id is 6506, canonical name is EnoughProjectives.
Accessed 1910 times total.
Classification:
| AMS MSC: | 18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories) | | | 18G05 (Category theory; homological algebra :: Homological algebra :: Projectives and injectives) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{P \ar[r]^p & A \ar[r] & 0}.$](http://images.planetmath.org:8080/cache/objects/6506/js/img1.png "$\displaystyle \xymatrix{P \ar[r]^p & A \ar[r] & 0}.$")