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projective object (Definition)

Let us consider the category of Abelian groups ${\bf Ab}_G$.

Definition 0.1   An object $P$ of an abelian category $\mathcal{A}$ is called projective if the functor $Hom_A (P,\^aˆ’) : \mathcal{A} \to {\bf Ab}_G$ is exact.

Remark.

This is equivalent to the following statement: An object $P$ is projective if given a short exact sequence $0 \to M\^a€² \to M \to M\^a€²\^a€² \to 0$ in an Abelian category $\mathcal{A}$, one has that:

$\displaystyle 0 \to Hom_{\mathcal{A}}(M\^a€², P) \to Hom_{\mathcal{A}}(M, P) \to Hom_{\mathcal{A}}(M\^a€²\^a€², P) \to 0$
is exact in ${\bf Ab}_G$.



"projective object" is owned by bci1.

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Keywords:  projective object

Cross-references: functor, abelian category, object, Abelian groups, category
There are 3 references to this object.

This is version 3 of projective object, born on 2009-06-15, modified 2009-06-15.
Object id is 800, canonical name is ProjectiveObject.
Accessed 425 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
None.
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