PlanetMath: projective object

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projective object

(Definition)

Let $\mathcal{C}$ be an abelian category. An object $P\in\mathcal{C}$ is called a projective object if

$\displaystyle \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Ab}$

is an exact functor, where $\mathbf{Ab}$ is the category of abelian groups.

The dual notion of a projective object is that of an injective object. An object in an abelian category $\mathcal{C}$ if the $\operatorname{Hom}(-,Q)$ functor from $\mathcal{C}$ to $\mathbf{Ab}$ is exact.

Examples. Let be a ring with 1. Consider the category of left -modules $\mathcal{M}_R$ . $\mathcal{M}_R$ is an abelian category. The projective objects in $\mathcal{M}_R$ are precisely the projective left -modules. So is itself a projective object in $\mathcal{M}_R$ . Dually, the injective objects in $\mathcal{M}_R$ are exactly the injective left -modules.

The definition given above can be generalized: an object in an arbitrary category $\mathcal{C}$ is projective if, given a diagram

$\displaystyle \xymatrix@+=4pc{ &{P}\ar[d]^{f}\ {A}\ar[r]_{g}&{B} }$

with

a strong epimorphism, there is a morphism $h:P\to A$ making the diagram below commutative

$\displaystyle \xymatrix@+=4pc{ &{P}\ar[d]^{f} \ar@{.>}[dl]_h \ {A}\ar[r]_{g}&{B} }$

Similarly, an injective object can be dually generalized.

Bibliography

1: F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)

"projective object" is owned by CWoo.

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Also defines:

injective object

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Cross-references: commutative, morphism, strong epimorphism, diagram, ring, functor, abelian groups, category, exact functor, object, abelian category
There are 6 references to this entry.

This is version 5 of projective object, born on 2004-11-01, modified 2008-09-22.
Object id is 6437, canonical name is ProjectiveObject.
Accessed 2204 times total.

Classification:

AMS MSC:

18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories)

Pending Errata and Addenda

None.

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